Realizable homotopy colimits (Q2927672)

Computing homotopy limits and colimits is a familiar subject investigated thoroughly within the framework of model categories. Nevertheless, if one is interested in the general case of a category \(\mathcal{C}\) endowed with a class \(\mathcal{W}\) of weak equivalences where a model structure is no longer to be seen, he or she cannot expect much at present. This paper is concerned with the question of computing homotopy limits and colimits in the more tractable case that \(\mathcal{W}\) is closed under coproducts. The main result of this paper is, as well as its converse, that the composition of the simple functor \(\boldsymbol{s}\) with the simplicial replacement of diagrams produces all homotopy colimits within the framework of \(\left( \mathcal{C},\mathcal{W}\right) \) abiding by such desired properties as cofinality, Fubini and preservation under forming diagram categories. It should be stressed that the homotopy colimits constructed in this way are stronger than the ones defined only at the level of localized categories, being really colimits in an appropriate \(2\)-category of categories with weak equivalences (or relative categories in the sense of [\textit{C. Barwick} and \textit{D. M. Kan}, Indag. Math., New Ser. 23, No. 1--2, 42--68 (2012; Zbl 1245.18006)]. The main result could be restated simply that to possess a simple functor is nothing else than to be homotopically cocomplete.