Pushouts of Dwyer maps are \((\infty, 1)\)-categorical
type: Item
instance of: scholarly article
title: Pushouts of Dwyer maps are \((\infty, 1)\)-categorical
zbMATH Open document ID: 1547.18032
author: Philip Hackney
author: Viktoriya Ozornova
author: Emily Riehl
author: Martina Rovelli
published in: Algebraic \& Geometric Topology
publication date: 2024-09-02T00:00:00Z
review text: Category theory has already fundamentally established itself as an efficient way to structure and compare different mathematical objects, including sets, groups or metric spaces. Unfortunately in some other areas of mathematics applying categorical methods faces serious challenges. For example, when studying topological spaces or chain complexes, we often consider them up to homotopy equivalence or chain equivalence, however, categories would only recover homeomorphisms or isomorphisms of chain complexes.\N\NHigher category theory, or more technically \((\infty,1)\)-category theory, arose as a response to this challenge. It has the benefit of being able to capture weak equivalences (e.g. homotopy equivalences or chain equivalences) in its definition and hence permits us to study such equivalences and their properties axiomatically. Hence, in recent decades \((\infty,1)\)-categories have become the standard framework for what is colloquially known as \emph{homotopical mathematics}: the parts of mathematics which naturally involve a notion of equivalence. While working in this framework comes with many benefits, it also has the significant drawback of involving many technical intricacies that one needs to contend with.\N\NOne key method to mitigate such intricacies is the \textit{nerve}, which is an embedding from categories into \((\infty,1)\)-categories. Whenever we can prove that the nerve preserves and/or reflects a property we can use that to our benefit to reduce \((\infty,1)\)-categorical computations to categorical ones. As a simple example, we know that the nerve preserves and reflects limits (as a fully faithful right adjoint), meaning in order to evaluate the limit of a diagram of categories inside \((\infty,1)\)-categories, we can simply evaluate the limit inside the category of categories and then apply the nerve.\N\NWhile this example has certain benefits, many crucial computations in \((\infty,1)\)-category theory involve pushouts. In general, a pushout…
reviewed by: Nima Rasekh
Mathematics Subject Classification ID: 18N60
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 7904852
zbMATH Keywords: Dwyer map
zbMATH Keywords: nerve functor
zbMATH Keywords: \((\infty, 1)\)-category
zbMATH Keywords: simplicial category
zbMATH Keywords: quasicategory
MaRDI profile type: Publication
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DOI: 10.2140/AGT.2024.24.2171
description: scientific article; zbMATH DE number 7904852