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A study of Kock's fat Delta

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arxiv 2503.10963 v2 pith:DLUQBR26 submitted 2025-03-14 math.CT math.AT

Tom de Jong , Nicolai Kraus , Simona Paoli , Stiéphen Pradal This is my paper
classification math.CTmath.AT
keywords deltacategorytheoryaritieskockmonadsnerveactive-inert
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Motivated by the study of weak identity structures in higher category theory we explore the fat Delta category, a modification of the simplex category introduced by J. Kock and provide a comprehensive study via the theory of monads with arities. Specifically, by proving that the free relative semicategory monad is strongly cartesian and identifying a dense generator, the theory of monads with arities immediately gives rise to the nerve theorem. We characterise the essential image of the nerve via the Segal condition, and show that fat Delta possesses an active-inert factorisation system. Building on these results, we also establish an isomorphism between two presentations of fat Delta.

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  1. On combinatorial aspects of fat Delta

    math.CT 2025-06 conditional novelty 7.0 of 10

    Kock's fat Delta category is generated by degenerated, vertical and standard faces subject to six relations, with every morphism factoring uniquely up to these relations.

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