Regular categories, oligomorphic monoids, and tensor categories
Abstract: Knop constructed a tensor category associated to a finitely-powered regular category equipped with a degree function. In recent work with Harman, we constructed a tensor category associated to an oligomorphic group equipped with a measure. In this paper, we explain how Knop's approach fits into our theory. The first, and most important, step describes finitely-powered regular categories in terms of oligomorphic monoids; this may be of independent interest. We go on to examine some aspects of this construction when the regular category one starts with is the category of $G$-sets for an oligomorphic group $G$, which yields some interesting examples.
- P. Deligne, J. Milne. Tannakian Categories. In “Hodge cycles, motives, and Shimura varieties,” Lecture Notes in Math., vol. 900, Springer–Verlag, 1982. DOI:10.1007/978-3-540-38955-2_4 Available at: http://www.jmilne.org/math/xnotes/tc.html
- Nate Harman, Andrew Snowden. Oligomorphic groups and tensor categories. arXiv:2204.04526
- Nate Harman, Andrew Snowden. Pre-Galois categories and Fraïssé’s theorem. arXiv:2301.13784
- Nate Harman, Andrew Snowden. Discrete pre-Tannakian categories. arXiv:2304.05375
- Friedrich Knop. Tensor envelopes of regular categories. Adv. Math. 214 (2007), pp. 571–617. DOI:10.1016/j.aim.2007.03.001 arXiv:math/0610552
- Sophie Kriz. Quantum Delannoy categories. Available at: https://krizsophie.github.io/
- Michael Megrelishvili, Menachem Shlossberg. Non-archimedean topological monoids. arXiv:2311.09187
- Andrew Snowden. Some fast-growing tensor categories. arXiv:2305.18230
- Andrew Snowden. On the representation theory of the symmetry group of the Cantor set. arXiv:2308.06648
- Andrew Snowden, Peter Webb. The biset category: a tensorial perspective. In preparation.
- Andrew Snowden, Peter Webb. The biset category: an oligomorphic perspective. In preparation.
- J. Wilson. The algebraic structure of ω𝜔\omegaitalic_ω-categorical groups. In “Groups—St. Andrews 1981,” eds. C. M. Campbell, E. F. Robertson, London Math. Soc. Lecture Notes 71 (1981), Cambridge, pp. 345–358. DOI:10.1017/CBO9780511661884
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