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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2-groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kan-fibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...

products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse intersection of submanifolds is

Abstract. Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring k. By ‘extend ’ we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme. The purpose of this paper is to show that there is a cyclic homology theory HC ∗ of schemes over a commutative ring k, extending the usual cyclic homology HC ∗ of k-algebras. By a cyclic homology theory for schemes over k we mean a family of graded k-modules HCn(X) associated to every scheme X over k which satisfy: (0.1) they are natural and contravariant in X; (0.2) for each affine scheme X = Spec A, there are natural isomorphisms HCn(X) ∼ = HCn(A) for all n; (0.3) if X = U ∪ V, there is a Mayer-Vietoris sequence · · · HCn(X) → HCn(U) ⊕ HCn(V) → HCn(U ∩ V) → HCn−1(X) · · ·. We discuss uniqueness of a cyclic homology theory briefly in Remark 0.5 below. We have chosen homological indexing because of axiom (0.2), and because cohomological indexing (HC n = HC−n) would concentrate the nonzero groups in negative degrees.

ix 0.1. Background and motivation ix 0.2. Subject matter of this book xvii

Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of G-valued singular cochains. Alternatively, one may regard H n (•, G) as a representable functor on the homotopy category of topological spaces, and thereby define H n rep(X, G) to be the set of homotopy classes of maps from X into an Eilenberg-MacLane space K(G, n). A third possibility is to use the sheaf cohomology H n sheaf (X, G) of X with coefficients in the constant sheaf G on X. If X is a sufficiently nice space (for example, a CW complex), then all three of these definitions agree. In general, however, all three give different answers. The singular cohomology of X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect H n sing (X, G) to tell us very much about X. Similarly, the cohomology group H n rep(X, G) is defined using maps from X into a simplicial complex, which (ultimately) relies on the existence of continuous real-valued functions on X. If X does not admit many real-valued functions, we should not expect H n rep (X, G) to be a useful invariant. However, the sheaf cohomology of X seems to be a good invariant for arbitrary spaces: it has excellent formal properties in general and sometimes yields

These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, July-August 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.

Abstract. Let G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the Lubin-Tate spectrum, X an arbitrary spectrum with trivial G-action, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous G-spectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is π∗( ( ˆ L(En∧X)) hG). We show that the homotopy fixed points of ˆ L(En ∧ X) come from the K(n)-localization of the homotopy fixed points of the spectrum (Fn ∧ X). 1.

In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. To explain the statement, recall that Baez-Dolan introduce the notion of k-uply monoidal n-category which is an n + k-category having only one i-morphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2-category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the n-category in question is an n-groupoid, this notion is—except for truncation at n—the same thing as the notion of k-fold iterated loop space, or “Ek-space ” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of k-uply monoidal n-categories for k ≫ n is what Grothendieck calls Picard n-categories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k ′ , the k-uply monoidal

decomposable topologies