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. 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.
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