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Simplicial presheaves. (English) Zbl 0624.18007
This paper is about closed model structures (in the sense of Quillen) on categories of simplicial presheaves (and sheaves). For the author, this means presheaves in the category of simplicial sets rather than simplicial objects in the category of presheaves; of course the two concepts coincide, but one’s point of view on this issue leads to the use of different tools in solving various problems which arise. The paper is well-written and motivated, and to a certain extent self-contained. It is well worth reading.
The central issue revolves around the fact that there are two natural closed model structures on the category of contravariant functors from \(\Delta\times\mathfrak C\) to \(\mathfrak{Sets}\). Specifically, focusing upon cofibrations \((= \) monomorphisms) leads to one structure – called by the author “global” – whereas taking fibrations \((= \) satisfy the Kan condition internally; for presheaves on a topological space this means that all of the maps on the stalks are [ordinary] fibrations) as the primary leads to another – “local” – theory. These are different theories, but because the map from a presheaf to its associated sheaf is a weak equivalence in either theory we have that the associated homotopy categories are equivalent. Among the people who have considered one or the other of these theories in the past are Brown, Gersten, Joyal, Thomason, and the reviewer.
What does the author add to earlier knowledge? Quite a bit, and I will not try to be exhaustive. Conceptually, his main contribution is in making clear that the topology on the site is to a certain extent incidental when one is considering the resulting homotopy theories. Beyond that, he has some nice applications of the theory to algebraic K-theory. There is a new description of etale K-theory, in terms of homotopy classes of maps from the terminal scheme to an iterated loop space. There is also a “descent-type” spectral sequence for morphisms in the simplicial etale topos. Generally, we are finding that more and more K-theories can be computed as cohomology of simplicial (pre)sheaves.

MSC:
18G55 Nonabelian homotopical algebra (MSC2010)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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