This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small

Version 2.0 March 16, 2008These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form.

Abstract. We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where ‘diagram category’ means diagrams of simplicial sets). This says that every combinatorial model

: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced group C -algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. Key words: Algebraic K and L-theory, Baum-Connes Conjecture, assembly maps, spaces and spectra over a category AMS-classification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left G-sets, and morphisms are G-maps. This is a useful construct for o...

Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents

Abstract. To any non-negatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian k-algebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence between Σg and the Deligne groupoid corresponding to g. The main result of the paper claims that the functor Σ commutes up to homotopy with the ”total space ” functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman [S1, S2, HS3] which implies that if a deformation problem is described “locally ” by a sheaf of dg Lie algebras g on a topological space X then the global deformation problem is described by the homotopy Lie algebra RΓ(X, g). 1.

Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3-sphere are combinatorially given. 1.

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The homotopy theoretic analogue of a compact Lie group is a p--ompact group, i.e a space X with finite mod--p cohomology and an loop structure given by an equivalence of the form X '\Omega BX. The `classifying space' BX has to be a p-complete space. We are concerned with the notions of centers and finite coverings of connected p--compact groups. In particular , we prove in this category two well known results for compact Lie groups; namely that the center of a connected p-compact group is finite iff the fundamental group is finite and that every connected p-compact group has a finite covering which is a product of a simply connected p-compact group and a torus. The latter statement also translates to connected finite loop spaces.

Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated