Pushouts of Dwyer maps are $(\infty,1)$categorical
Abstract
The inclusion of 1categories 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 previouslyidentified class of functors, then the 1categorical 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 1categories. 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 1categorical pushouts are $(\infty,1)$categorical.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.02353
 arXiv:
 arXiv:2205.02353
 Bibcode:
 2022arXiv220502353H
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 18N60;
 55U35
 EPrint:
 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