num_pages: 944
description: Higher category theory is generally regarded as technical and forbidding but part of it is considerably more the theory of infinity-categories higher categories in which all higher morphisms are assumed to be invertible. In Higher 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.
language: English
title: Higher Topos Theory (AM-170)
publisher: Princeton University Press