Pushouts of Dwyer maps are $(\infty ,1)$-categorical

Abstract

The inclusion of 1-categories into $(\infty ,1)$-categories fails to preserve colimits in general, and pushouts in particular. In this note, we observe that if one functor in a span of categories belongs to a certain previously-identified class of functors, then the 1-categorical pushout is preserved under this inclusion. Dwyer maps, a kind of neighborhood deformation retract of categories, were used by Thomason in the construction of his model structure on 1-categories. Thomason previously observed that the nerves of such pushouts have the correct weak homotopy type. We refine this result and show that the weak homotopical equivalence is a weak categorical equivalence. We also identify a more general class of functors along which 1-categorical pushouts are $(\infty ,1)$-categorical.

v3: Major rewrite based on an alternate proof strategy proposed by a referee that uses a different model of $\infty$-categories. 12 pages. Prior proof may be found in v2. v2: Minor clarifications and corrections suggested by a referee. v1: An expansion and correction of a result from arxiv:2106.03660v2

References in corpus (2)

• An $(\infty ,2)$-categorical pasting theorem
• A counterexample in quasi-category theory