Quillen closed model structures for sheaves
Suppose that \({\mathcal C}\) is a category with a Quillen closed model structure and that there is a pair of functors \(L :{\mathcal C}\to{\mathcal D}\) and \(R :{\mathcal D}\to{\mathcal C}\) such that \(L\) is left adjoint to \(R\). Under fairly weak conditions, it is shown that \(D\) has an induced Quillen closed model structure. The result is applied with \({\mathcal C}\) the category of simplicial sheaves on a fixed site and \({\mathcal D}\) the category of sheaves of 2-groupoids, or of bisimplicial sheaves, or of simplicial sheaves of groupoids. In the first case, the homotopy category of \({\mathcal D}\) is equivalent to the full subcategory of the homotopy category of \({\mathcal C}\) generated by objects \(X\) such that \(\pi_nX = 0\) for \(n > 2\). In the remaining two cases, the homotopy category of \({\mathcal D}\) is equivalent to that of \({\mathcal C}\).
scientific article; zbMATH DE number 797257
scientific article