Cats

Abstract: A generalization of the notion of an $\infty$-category is presented, allowing for ($\infty$-)cat(egorie)s that may have non-invertible higher morphisms.

• The paper introduces "cats" as a generalization of infinity-categories that accommodates non-invertible higher morphisms, based on the category of generalized simplices.
• Key conjectures explore the equivalence of model structures generated by spines and inner horns, asserting that fibrant objects represent these "cats" satisfying specific horn-filling conditions.
• The research delves into the definitions of faces, horns, and spines, contrasts "cats" with groupoids, and touches upon connections to $n$-categories and structures related to group cohomology.

The paper "Cats" by Daniel Gerigk explores a generalization of the concept of an $\infty$-category, introducing structures that accommodate non-invertible higher morphisms. The study begins by examining the category of generalized simplices, denoted as $A$, which was originally introduced by Simpson under the notation $\Theta$. The research investigates the properties of simplicial objects, particularly focusing on those termed "cats," where every inner horn can be filled, implying that these objects satisfy specific horn-filling conditions.

Key conjectures are formulated in this framework, prominently regarding the equivalence of the model structures generated by spines and inner horns. The generalized simplicial category plays a crucial role in establishing these conjectures. The author hypothesizes that every spine is inner anodyne and that the Cisinski model structure generated by spines is equivalent to that generated by inner horns. These conjectures are vital in asserting that fibrant objects within this model structure encapsulate all "cats."

The terminology employed includes definitions of faces, horns, and spines, with particular attention to inner structures crucial for defining the morphisms with desired properties. The research extends to notions of strictness in cats and groupoids, emphasizing structures where maps align with equivalences typically seen in well-defined categorical contexts.

For $n$-categories, a sequence of embeddings from $_n$ to the complete category $\Set[]$ is developed, illustrating the nested nature of these structures as dimensions increase. Here, $n$-cats demonstrate properties distinct from higher-dimensional cats, yet important for understanding complex categorical structures at different levels.

The paper also delineates the model structure for groupoids, differentiated from cats by the usage of boundary mappings $A \to 1$, highlighting the variances in fibrant objects across different categorical frameworks.

Finally, within the context of $H^2(G; A)$, connections between group structures are articulated. Definitions for structures like $\bfB^1 G$ and $\rmB^2 A$ are provided to comprehend associations between group maps and cocycles in higher categorical settings.

The paper is theoretical, centering around definitions, conjectures, and structural assessments, with implications for categorical algebra primarily in homotopy theory and higher category theory. It serves as groundwork for further exploration into non-standard categorical constructs that defy traditional invertibility constraints.