\(A\)-cellular homotopy theories
Given a suitable pointed closed simplicial model category \({\mathcal C}\) and a cofibrant object \(A\) in \({\mathcal C}\), the author proves the existence of a new closed model category structure in \({\mathcal C}\) in which cofibrant objects are built by attaching ``\(A\)-cells''. Especially, if \({\mathcal C}\) is the category of pointed spaces the \(A\)-cells are half-smash products \(A\rtimes \Delta[n]\), and the \(A\)-cellular structure coincides with the usual CW-structure when \(A=S^0\). The author uses this existence theorem in order to prove that there exist \(A\)-cellular closed simplicial model category structures in the following categories: (1) in the category of pointed spaces with respect to a cofibrant space \(A\), (2) in the category of pointed simplicial sets with respect to a pointed simplicial set \(A\), (3) in the category of simplicial groups with respect to a cofibrant simplicial group \(A\), (4) in the category of diagrams of pointed spaces over a small category \({\mathcal I}\) with respect to \(A\wedge({\mathcal I}/-)_+\), where \(({\mathcal I}/-)_+\) is the overcategory with a disjoint base point, (5) in the category of pointed simplicial sets over a small category \({\mathcal I}\) with respect to the diagram \(A\wedge({\mathcal I}/-)_+\), (6) in the category of diagrams of simplicial groups over a small category \({\mathcal I}\) with respect to \(A\otimes ({\mathcal I}/-)\), where \(A\) is a cofibrant simplicial group. For diagrams over a contractible category the closed model category structures have the property that under suitable conditions the homotopy limit functor commutes with a factorization of a map into a cofibration followed by a trivial fibration up to an equivalence.
scientific article; zbMATH DE number 1340438