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Simplicial objects in a Grothendieck topos. (English) Zbl 0606.18006
Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 193-239 (1986).
[For the entire collection see Zbl 0588.00014.]
A simplicial homotopy is defined in the context of a Grothendieck topos. The result generalizes the work of K. S. Brown concerning simplicial sheaves on a topological space [Trans. Am. Math. Soc. 186 (1973), 419-458 (1974; Zbl 0245.55007)].
The fibrations in the category of simplicial sheaves on an arbitrary Grothendieck site are defined by local lifting property, the weak equivalences, via local homotopy group sheaves isomorphisms.
Via the homotopy category, cohomology groups are defined. Between other results, coincidence with étale cohomology is given. This paper trends towards a study of algebraic groups over an algebraically closed field and proves an isomorphism conjecture on G.
Reviewer: G.Hoff

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18F10 Grothendieck topologies and Grothendieck topoi
14F20 Étale and other Grothendieck topologies and (co)homologies
14F35 Homotopy theory and fundamental groups in algebraic geometry
20G99 Linear algebraic groups and related topics
55U35 Abstract and axiomatic homotopy theory in algebraic topology