eBook

Higher topos theory

Lurie, Jacob

2009

Overview

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.

• Publisher: Princeton University Press
• Subject: Adjoint functors / Canonical map / Categories (Mathematics) / Category of sets / Category theory / Coequalizer / Cofinality / Coherence theorem / Cohomology / Cokernel / Commutative property / Contractible space / Coproduct / Corollary / CW complex / Derived category / Diagonal functor / Diagram (category theory) / Enriched category / Equivalence class / Equivalence relation / Existence theorem / Existential quantification / Functor / Functor category / Grothendieck topology / Grothendieck universe / Groupoid / Higher category theory / Higher Topos Theory / Homotopy / Homotopy category / Homotopy colimit / Homotopy group / Inclusion map / Kan extension / Limit (category theory) / MATHEMATICS / MATHEMATICS / Algebra / General / MATHEMATICS / Geometry / General / Maximal element / Metric space / Model category / Monoidal category / Monoidal functor / Monomorphism / Morphism / Natural transformation / O-minimal theory / Open set / Presheaf (category theory) / Pullback (category theory) / Pushout (category theory) / Quillen adjunction / Quotient by an equivalence relation / Retract / Right inverse / Sheaf (mathematics) / Sheaf cohomology / Simplicial category / Simplicial set / Special case / Subcategory / Surjective function / Theorem / Topological space / Topology / Topos / Toposes / Total order / Transitive relation / Upper and lower bounds / Weak equivalence (homotopy theory) / Yoneda lemma / Zorn's lemma
• ISBN: 9780691140490, 1400830559, 9781400830558, 0691140480, 0691140499, 9780691140483