2025/10/05 by Kimball Strong, Strong, Kimball
Computer Science · #18N30 (Primary) 18N60 (Secondary) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2510.04254
openalex publication_date 2025/10/05 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28
We define a functor which takes in an (∞,1)-category and outputs an (ω,1)-category, the natural maximally "strict" version of an (∞,1)-category. We do this by modeling (∞,1)-categories as categories enriched in ∞-groupoids, and then "locally strictifying" (applying the strictification of ∞-groupoids to each hom space) to obtain a category enriched in ω-groupoids with respect to the Gray tensor product, followed by "globally strictifying" (strictifying the enrichment from the Gray tensor product to the cartesian product) to obtain a category cartesian-enriched in ω-groupoids, which is equivalently an (ω,1)-category. We prove that this functor is conservative by proving a slightly stronger statement on systems of chain complexes parameterized by the homotopy (2,1)-category of an (∞,1)-category, and explain how this generalizes the Homological Whitehead Theorem from spaces to (∞,1)-categories.