Homotopy theory of simplicial sheaves in completely decomposable topologies.

*(English)*Zbl 1194.55020The homotopy theory of simplicial (pre)sheaves introduced by Joyal and developed by J. F. Jardine [J. Pure Appl. Algebra 47, 35–87 (1987; Zbl 0624.18007), Can. J. Math. 39, 733–747 (1987; Zbl 0645.18006)] provides us with a proper simplicial model structure on the category of simplicial sheaves and presheaves and allows us to give definitions of all the important objects of homotopical algebra in the context of sheaves on all sites. As a consequence of its generality, some of the important classes of objects and morphisms have a very abstract definition. On the other hand, the Brown-Gersten approach [K. S. Brown and S. M. Gersten, Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 266–292 (1973; Zbl 0291.18017)] for simplicial sheaves on Noetherian topological spaces of finite dimension is much more explicit, and gives a finitely generated model structure on the category of simplicial sheaves, but does not generalize to arbitrary sites.

The author of the paper under review defines a class of sites for which a generalized analog of the Brown-Gersten approach works. He proves that for such sites the class of weak equivalences of simplicial presheaves can be generated by an explicitly given set of generating weak equivalences.

The author of the paper under review defines a class of sites for which a generalized analog of the Brown-Gersten approach works. He proves that for such sites the class of weak equivalences of simplicial presheaves can be generated by an explicitly given set of generating weak equivalences.

Reviewer: J. Remedios (La Laguna)

##### MSC:

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

54B40 | Presheaves and sheaves in general topology |

##### Keywords:

simplicial sheaves##### References:

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