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The bifiltration of a relation and extended Dowker duality

Published 17 Oct 2023 in math.AT | (2310.11529v1)

Abstract: We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent spaces, thereby extending Dowker's duality theorem. We also prove a functorial version of this result. Specializing the above construction a bifiltration of Dowker complexes that coherently incorporates the total weights of a relation's row and column complex into one single object is introduced. This construction is motivated by challenges in data analysis that necessitate the simultaneous study of a data matrix rows and columns. To illustrate the applicability of our constructions for solving those challenges we give an appropriate reconstruction result.

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References (19)
  1. Björner, A.: Topological methods. Handbook of combinatorics 2, 1819–1872 (1995) Chowdhury and Mémoli [2018] Chowdhury, S., Mémoli, F.: A functorial dowker theorem and persistent homology of asymmetric networks. Journal of Applied and Computational Topology 2, 115–175 (2018) Virk [2021] Virk, Ž.: Rips complexes as nerves and a functorial dowker-nerve diagram. Mediterranean Journal of Mathematics 18(2), 1–24 (2021) Robinson [2022] Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Chowdhury, S., Mémoli, F.: A functorial dowker theorem and persistent homology of asymmetric networks. Journal of Applied and Computational Topology 2, 115–175 (2018) Virk [2021] Virk, Ž.: Rips complexes as nerves and a functorial dowker-nerve diagram. Mediterranean Journal of Mathematics 18(2), 1–24 (2021) Robinson [2022] Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Virk, Ž.: Rips complexes as nerves and a functorial dowker-nerve diagram. Mediterranean Journal of Mathematics 18(2), 1–24 (2021) Robinson [2022] Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  2. Chowdhury, S., Mémoli, F.: A functorial dowker theorem and persistent homology of asymmetric networks. Journal of Applied and Computational Topology 2, 115–175 (2018) Virk [2021] Virk, Ž.: Rips complexes as nerves and a functorial dowker-nerve diagram. Mediterranean Journal of Mathematics 18(2), 1–24 (2021) Robinson [2022] Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Virk, Ž.: Rips complexes as nerves and a functorial dowker-nerve diagram. Mediterranean Journal of Mathematics 18(2), 1–24 (2021) Robinson [2022] Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  3. Virk, Ž.: Rips complexes as nerves and a functorial dowker-nerve diagram. Mediterranean Journal of Mathematics 18(2), 1–24 (2021) Robinson [2022] Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  4. Robinson, M.: Cosheaf representations of relations and dowker complexes. Journal of Applied and Computational Topology 6(1), 27–63 (2022) Segal [1968] Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  5. Segal, G.: Classifying spaces and spectral sequences. Publications Mathématiques de l’IHÉS 34, 105–112 (1968) Dugger and Isaksen [2004] Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  6. Dugger, D., Isaksen, D.C.: Topological hypercovers and 1-realizations. Mathematische Zeitschrift 246(4), 667–689 (2004) Curto and Itskov [2008] Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  7. Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS computational biology 4(10), 1000205 (2008) Singh et al. [2008] Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  8. Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. Journal of vision 8(8), 11–11 (2008) Rybakken et al. [2019] Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  9. Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological feature extraction. Neural computation 31(1), 68–93 (2019) Gardner et al. [2022] Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  10. Gardner, R.J., Hermansen, E., Pachitariu, M., Burak, Y., Baas, N.A., Dunn, B.A., Moser, M.-B., Moser, E.I.: Toroidal topology of population activity in grid cells. Nature 602(7895), 123–128 (2022) Vaupel et al. [2023] Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  11. Vaupel, M., Erik, H., Dunn, B.: A topological perspective on the dual nature of the correlation structure and the neural state space. bioRxiv preprint (2023) Friedman [2008] Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  12. Friedman, G.: An elementary illustrated introduction to simplicial sets. arXiv preprint arXiv:0809.4221 (2008) Goerss and Jardine [2009] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  13. Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Springer, Berlin (2009) Quillen [1973] Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  14. Quillen, D.: Higher algebraic k-theory: I. In: Higher K-theories, pp. 85–147. Springer, Berlin (1973) Waldhausen [1982] Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  15. Waldhausen, F.: Algebraic k-theory of spaces, a manifold approach. In: Current Trends in Algebraic Topology, Part 1, pp. 141–184 (1982) Segal [1974] Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  16. Segal, G.: Categories and cohomology theories. Topology 13(3), 293–312 (1974) The RIVET Developers [2020] The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  17. The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ Dugger et al. [2004] Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  18. Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51 (2004). Cambridge University Press Freund et al. [2015] Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015) Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
  19. Freund, A., Andreatta, M., Giavitto, J.-L.: Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence 73, 311–334 (2015)
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  1. Melvin Vaupel 
  2. Benjamin Dunn 

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