Pushouts of Dwyer maps are $(\infty,1)$-categorical
Abstract
The inclusion of 1-categories into -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 -categorical.
Keywords
Cite
@article{arxiv.2205.02353,
title = {Pushouts of Dwyer maps are $(\infty,1)$-categorical},
author = {Philip Hackney and Viktoriya Ozornova and Emily Riehl and Martina Rovelli},
journal= {arXiv preprint arXiv:2205.02353},
year = {2024}
}
Comments
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