Homotopy theory of diagrams (Q2784248)

The fundamental concept of this clearly written paper which puts a special emphasis on the construction of homotopy colimits and limits in arbitrary model categories in the sense of \textit{D. G. Quillen} [``Homotopical algebra,'' Lect. Notes Math. 43 (1967; Zbl 0168.20903)] is that of a (left) model approximation for a category \({\mathcal C}\) with weak equivalences, i.e., \({\mathcal C}\) is equipped with a class of morphisms (weak equivalences) satisfying the `two of three' property. A (left) model approximation \({\mathcal M}\) of \({\mathcal C}\) is a model category \({\mathcal M}\) together with a pair of adjoint functors NEWLINE\[NEWLINE\begin{matrix} & l\\ {\mathcal M} & \rightleftarrows & {\mathcal C}\\ & r \end{matrix}NEWLINE\]NEWLINE where \(l\dashv r\), satisfying certain properties. For example, the category of simplicial sets together with the realization and the singular functor is a left model approximation of the category of CW-complexes. The key result says that if \({\mathcal C}\) admits a model approximation, then so does the functor category \({\mathcal F}un(I,{\mathcal C})\). It is shown that from the homotopy theoretical point of view being a model category or having a model approximation does not make much difference. In both cases, amongst other things, one can form the localized homotopy category, construct suspensions and general homotopy colimits from Puppe sequences, and construct mapping spaces.