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Quillen closed model structures for sheaves. (English) Zbl 0828.18005

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 \({\mathcal C}\).

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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References:

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