Explore the vast range of titles available.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. InHigher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

We give a new functorial construction of the space of triangles introduced by Le Barz. This description is used to exhibit the space as a composition of smooth blowups, to obtain a space of unordered triangles, and to study how the space varies in a family.

Given a pair of short exact sequences $\\left. 1 \\right)0 \\to X_ \\to ^\\gamma Y_ \\to ^\\delta Z \\to 0,0 \\to A_ \\to ^\\alpha B_ \\to ^\\beta C \\to 0$ in an abelian category A̰, with sufficiently many projectives and injectives, and given an additive bifunctor T we show that T applied to the pair (1) gives rise to a diagram of a type described by C. T. C. Wall that contains 15 interlocking long exact sequences involving the derived functors of T at (A, X), (A, Y), etc. and also involving the derived functors of TP and TQ which are two functors with domain A̰² that arise through the failure of T to preserve pullbacks and pushouts. In the case of Hom (respectively ⨂) in the category of G–modules for a group G the derived functors of TP (respectively TQ) are expressed in terms of group cohomology (respectively homology).

Previous extensions of degree theory to multi-valued mappings have required convexity or acyclicity conditions on the domains or point images of the mapping being considered. By using a straightforward combination of the results of D. G. Bourgin with results of K. Geba and A. Granas, a degree is defined in this paper which removes the acyclicity conditions, provided that the point images are acyclic in high enough dimensions. Using the degree, some fixed point theorems are developed.