Une remarque sur le classifiant d'une catégorie topologique. (A remark on the classifying space of a topological category) (Q1082649)

The author answers a question of R. Brown concerning pushouts (amalgamated sums) of topological categories. The main result is as follows. Let \({\mathcal F}\) be a topological category with objects X and \(f: X\to Y\) a map. Consider X and Y as topological categories. Then the pushout \({\mathcal G}\) of the diagram \({\mathcal F}\leftarrow X\to Y\) has a classifying space \({\mathcal B}{\mathcal G}\) which is weakly homotopy equivalent to the pushout of B\({\mathcal F}\leftarrow BX\to BY\), that is, B and \({\mathcal G}\) commute on such diagrams. The author works in the category of k-spaces and for the proof requires that certain natural inclusions in the nerve of F be cofibrations or h- cofibrations. This condition T is observed to be natural in several concrete cases. The main technical ingredient of the proof is a specific construction of the pushout category \({\mathcal G}\).