A study of Kock's fat Delta

TL;DR

This work studies J. Kock's fat Delta category $\underline{\Delta}$, a modification of the simplex category designed to weaken unit identities in higher categories and to facilitate diagrammatic interpretations of weak identities in both higher category theory and type theory. The authors develop the theory of monads with arities to analyze $\underline{\Delta}$, constructing the free semicategory and the free relative semicategory monads $\mathtt{f}$ and $\mathtt{f}^{\mathsf{r}}$, and proving these monads are strongly cartesian. They identify the arities of the free relative semicategory monad with $\underline{\Delta}_0$ and prove an isomorphism between two presentations of fat Delta, thereby establishing a nerve theorem: the nerve functor from relative semicategories to presheaves on $\underline{\Delta}$ is fully faithful and its essential image is characterized by a Segal condition. Consequently, $\underline{\Delta}$ is a hypermoment category that is strongly unital and extensional, with an active–inert factorisation system and a dense Segal core, providing a robust framework for higher operadic and homotopy-theoretic applications within both category theory and homotopy type theory.

Abstract

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. We provide a comprehensive study of fat Delta via the theory of monads with arities, and use these results to show that fat Delta is a hypermoment category in the sense of C. Berger. 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 and show that it is a strongly unital and extensional hypermoment category.

Figures (1)

• Figure 1: Initial segments of the categories $\Delta$ (left) and its modification $\underline{\Delta}$ (right) by Kock Kock2006. While $\Delta$ is a Reedy category (having positive and negative morphisms which go "up" and "down"), $\underline{\Delta}$ is a direct category (its morphisms only go "up"). The objects of $\underline{\Delta}$ are the epimorphisms of $\Delta$: the top row of $\underline{\Delta}$ are the identities, the second row shows the epimorphisms of $\Delta$ that are drawn in the diagram on the left, and the third row represents $[2] \twoheadrightarrow [0]$.

Theorems & Definitions (75)

• Theorem 1: \ref{['thm:fDeldesc']}
• Theorem 2: \ref{['thm:fnerve']}
• Theorem 3: \ref{['thm:hypcatfDel']}
• Remark 2.1
• Lemma 2.2: Leinster2004
• Proposition 2.3
• proof
• Proposition 2.4: BMW2012
• Proposition 2.5
• proof