Presentation of finite Reedy categories as localizations of finite direct categories

Genki Sato

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arXiv · 2502.05096

Presentation of finite Reedy categories as localizations of finite direct categories

Genki Sato

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Abstract

In this paper, we present a construction from a Reedy category $C$ of a direct category $\operatorname{Down}(C)$ and a functor $\operatorname{Down}(C) \to C$, which exhibits $C$ as an $(\infty,1)$-categorical localization of $\operatorname{Down}(C)$. This result refines previous constructions in the literature by ensuring finiteness of the direct category $\operatorname{Down}(C)$ whenever $C$ is finite, which is not guaranteed by existing approaches. The finiteness property is useful when we want to embed the construction into the syntax of a (non-infinitary) logic: the author expects the construction may be used to develop a meta-theory of finitely truncated simplicial types for homotopy type theory.

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BibTeX RIS

Genki Sato. 2025-02-07. Presentation of finite Reedy categories as localizations of finite direct categories. https://arxiv.org/abs/2502.05096

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