Higher Topos Theory
Statement
Higher Topos Theory develops the theory of ∞-topoi: higher-categorical generalizations of Grothendieck toposes in which objects behave like sheaves of spaces (homotopy types) rather than sheaves of sets. Using quasi-categories (weak Kan complexes) and the Joyal model structure, Lurie axiomatizes presentable ∞-categories, accessible localizations, and sheaves on Grothendieck sites in this homotopy-coherent setting. The framework subsumes ordinary 1-topos theory, provides a model-independent basis for derived algebraic geometry (spectral schemes, derived stacks), and aligns with homotopy type theory / univalent foundations at the level of conceptual semantics. The monograph is a systematic body of definitions, constructions, and proved theorems in higher category theory; formalizable in standard set-theoretic foundations (ZFC) once the underlying ∞-categorical objects are encoded (e.g. via simplicial sets with appropriate model-categorical or type-theoretic semantics).
Formula
∞-topos ≃ left-exact accessible localization of PSh(C, 𝒮) (presentable ∞-categories)
Equation
Quasi-categories; Cartesian fibrations; Grothendieck construction
Proof sketch
Lurie, J. Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, 2009. Builds on Joyal’s quasi-categories and Rezk’s complete Segal spaces; internal development of limits, adjunctions, and sheafification in the ∞-categorical setting.
Source
Lurie, J. Higher Topos Theory. Annals of Mathematics Studies 170, Princeton University Press, 2009. Available as arXiv:math/0608040.
Domains
mathematics, logic