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arXiv · 2205.02353

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

Philip Hackney·Viktoriya Ozornova·Emily Riehl·Martina Rovelli
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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.

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Philip Hackney, Viktoriya Ozornova, Emily Riehl, Martina Rovelli. 2022-05-04. Pushouts of Dwyer maps are $(\infty,1)$-categorical. https://doi.org/10.2140/agt.2024.24.2171

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