An Enriched Approach to the Strictification of $(\infty,1)$-Categories

Abstract

We define a functor which takes in an $(\infty,1)$-category and outputs an $(\omega,1)$-category, the natural maximally "strict" version of an $(\infty,1)$-category. We do this by modeling $(\infty,1)$-categories as categories enriched in $\infty$-groupoids, and then "locally strictifying" (applying the strictification of $\infty$-groupoids to each hom space) to obtain a category enriched in $\omega$-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 $\omega$-groupoids, which is equivalently an $(\omega,1)$-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 $(\infty,1)$-category, and explain how this generalizes the Homological Whitehead Theorem from spaces to $(\infty,1)$-categories.