Brown categories and bicategories (Q505356)

Given a category with specified weak equivalences, there is a simplicial localization construction of \textit{W. G. Dwyer} and \textit{D. M. Kan} [J. Pure Appl. Algebra 18, 17--35 (1980; Zbl 0485.18013)] which produces a simplicially enriched category. However, there are now many other known equivalent structures to simplicially enriched categories, such as Segal categories and complete Segal spaces. The goal of this paper is to construct a Segal category associated to a Brown category (or category of cofibrant objects) in a way that is compatible with the Dwyer-Kan simplicial localization. In fact, the author considers a weaker version of a Brown category, which he calls a partial Brown category, which possesses only the structure needed for this construction to work. The methods used in the paper include the classifying diagram construction defined by Rezk and used by Barwick and Kan to establish a Quillen equivalence between relative categories and complete Segal spaces.