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

TL;DR

This paper investigates conditions under which 1-categorical pushouts are preserved in the transition to $( abla,1)$-categories, focusing on Dwyer maps and their generalizations, and refines the understanding of their homotopical properties.

Contribution

It identifies that Dwyer maps and certain functors preserve 1-categorical pushouts in the $( abla,1)$-categorical setting and refines their homotopical equivalence to a weak categorical equivalence.

Findings

Dwyer maps preserve pushouts in the $( abla,1)$-categorical context.

Nerves of these pushouts have the correct weak homotopy type.

Weak homotopical equivalence is a weak categorical equivalence.

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.