Pushouts of Dwyer maps are \((\infty, 1)\)-categorical
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 of \((\infty,1)\)-categories can be significantly more complicated than the initial input. In particular it is not the case that the nerve preserves all pushouts. Given this situation, a key task in the study of \((\infty,1)\)-category theory is to recognize sufficient conditions a pushout diagram of categories needs to satisfy so that it is preserved via the nerve.\N\NIn this paper the authors precisely tackle this question. They consider the notion of a \textit{Dwyer map}, originally introduced in [\textit{R. W. Thomason}, Cah. Topologie Géom. Différ. Catégoriques 21, 305--324 (1980; Zbl 0473.18012)], and prove in \textit{Theorem 1.5} that pushouts of categories where one leg is a Dwyer map are preserved via the nerve, meaning they result in a pushout diagram of \((\infty,1)\)-categories. As an application, they use this result in a follow up paper to prove the uniqueness of the composition of pasting diagrams in an \((\infty,2)\)-category [\textit{P. Hackney} et al., Trans. Am. Math. Soc. 376, No. 1, 555--597 (2023; Zbl 1505.18031)].
scientific article; zbMATH DE number 7904852
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