Higher Topos Theory

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

• Publication year

  2006

• Venue

  Unknown venue

• Publication date

  2006-08-02

• Fields of study

  Mathematics

• Identifiers
  arXiv math/0608040
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Sur quelques points d'algebre homologique

  2009cited by this paper
• Simplicial Homotopy Theory

  2009influential reference
• Ordinal subdivision and special pasting in quasicategories

  2008cited by this paper
• On quasi-categories as a foundation for higher algebraic stacks

  2007cited by this paper
• Vers une axiomatisation de la théorie des catégories supérieures.

  2005influential reference
• Homotopy Limit Functors on Model Categories and Homotopical Categories

  2005cited by this paper
• Rigidification of algebras over multi-sorted theories

  2005cited by this paper
• A model category structure on the category of simplicial categories

  2004cited by this paper
• Parametrized homotopy theory

  2004cited by this paper
• Higher Monodromy

  2004cited by this paper
• Rigid analytic geometry and its applications

  2003cited by this paper
• Model categories and their localizations

  2003influential reference
• Homotopical algebraic geometry. I. Topos theory.

  2002cited by this paper
• Hypercovers and simplicial presheaves

  2002influential reference
• Vers une interprétation galoisienne de la théorie de l'homotopie

  2002cited by this paper
• Quasi-categories and Kan complexes

  2002influential reference
• A Survey of Definitions of n-Category

  2001cited by this paper
• Simplicial structures on model categories and functors

  2001cited by this paper
• Proper Maps of Toposes

  2000cited by this paper
• Combinatorial Model Categories Have Presentations

  2000influential reference
• A model for the homotopy theory of homotopy theory

  1998cited by this paper
• Tame Topology and O-minimal Structures

  1998cited by this paper
• Homotopy Coherent Category Theory

  1996cited by this paper
• Coherence of tricategories

  1995cited by this paper
• On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets

  1995cited by this paper
• On the classification of 2-gerbes and 2-stacks

  1994cited by this paper
• Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture

  1994cited by this paper
• Simplicial objects in algebraic topology

  1993cited by this paper
• Sheaves In Geometry And Logic

  1992cited by this paper
• Spectral Theory and Analytic Geometry over Non-Archimedean Fields

  1990cited by this paper
• Sheaves on Manifolds

  1990cited by this paper
• The Finiteness Obstruction of C. T. C. Wall

  1989cited by this paper
• Resolutions of unbounded complexes

  1988cited by this paper
• Etale Cohomology and the Weil Conjecture

  1988cited by this paper
• Realizing diagrams in the homotopy category by means of diagrams of simplicial sets

  1984cited by this paper
• Etale Homotopy of Simplicial Schemes.

  1982cited by this paper
• Sur la notion de diagramme homotopiquement cohérent

  1982cited by this paper
• Two-dimensional sheaf theory

  1982cited by this paper
• Simplicial localizations of categories

  1980cited by this paper
• Lectures on Hilbert Cube Manifolds

  1976cited by this paper
• Cech and Steenrod Homotopy Theories with Applications to Geometric Topology

  1976cited by this paper
• The localization of spectra with respect to homology

  1975cited by this paper
• Mappings between ANRs that are fine homotopy equivalences

  1975cited by this paper
• Abstract homotopy theory and generalized sheaf cohomology

  1973cited by this paper
• Théorie des Topos et Cohomologie Etale des Schémas

  1972influential reference
• Cohomologie non abélienne

  1971influential reference
• Classifying spaces and spectral sequences

  1968cited by this paper
• Finiteness conditions for CW-complex I

  1965cited by this paper
• Homotopy associativity of $H$-spaces. II

  1963cited by this paper
• Sur quelques points d'algèbre homologique, I

  1957cited by this paper
• Axiomatic Approach to Homology Theory.

  1945cited by this paper

• Factorization algebras in quite a lot of generality

  2026cites this paper
• Deformation Theory of $\mathbb{E}_n$-Monoidal Categories

  2026cites this paper
• Derived algebras on formal stacks and prismatic gauges

  2026influential citation
• The Homology of Complex Equivariant Bordism

  2026cites this paper
• Modeling (∞,1)$(\infty,1)$ ‐categories with Segal spaces

  2026influential citation
• On the Kelly monoidal structure of $\Lambda$-sequences and unital operads

  2026cites this paper
• The Riemann Hypothesis: Past, Present and a Letter Through Time

  2026cites this paper
• Deformation theory and cotangent complex of dg operads

  2026cites this paper
• The equifibered approach to ∞-properads

  2026cites this paper
• The Torsion of Automorphisms of Nilpotent Spaces

  2026influential citation
• A \v{C}ech--Stokes Pushout Groupoid: a Log/Kummer Betti Presenter for Stokes Torsors

  2026influential citation
• On the Farrell–Jones conjecture for localising invariants

  2026cites this paper
• Witt affine Springer theory

  2026cites this paper
• The $\infty$-category of $\infty$-categories in simplicial type theory

  2026cites this paper
• D\'evissage for Algebraic K-theory of Small Stable $\infty$-categories

  2026cites this paper
• On Dual Algebras of Hopf Algebroids

  2026cites this paper
• Classifying anima of condensed $\infty$-categories of points

  2026cites this paper
• Ganea decompositions of classifying spaces

  2026influential citation
• On geometric models in representation theory

  2026cites this paper
• Quasi-unitial Inner Kan Spaces

  2026influential citation
• Consciousness in a Higher Categorical Context

  2026cites this paper
• Topological CoHochschild Homology and Thom Spectra

  2026influential citation
• On Gluing Data, Finite Ringed Spaces and schemes

  2026cites this paper
• The Bass–Quillen Conjecture for Torsors Over Valuation Rings

  2025cites this paper
• On duoidal ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories

  2025influential citation
• Symmetries of the cyclic nerve

  2025cites this paper
• The Hilton–Milnor theorem in higher topoi

  2025influential citation
• Universality of Barwick's unfurling construction

  2025cites this paper
• On endomorphisms of topological Hochschild homology

  2025cites this paper
• The Derived Adelic Cohomology Conjecture for Elliptic Curves

  2025cites this paper
• The Yoneda embedding in simplicial type theory

  2025cites this paper
• Non-commutative nature of ℓ-adic vanishing cycles

  2025cites this paper
• The quantum torus as an $\mathbb E_M$-category

  2025cites this paper
• $(-1)$-Shifted Darboux theorem of derived schemes in characteristic $p>2$

  2025cites this paper
• Abstract representation theory via coherent Auslander-Reiten diagrams

  2025influential citation
• $G$-gerbes on perfectoid spaces

  2025influential citation
• The Parabolic K-motivic Hecke Category

  2025cites this paper
• A note on ideals in derived geometries

  2025cites this paper
• Geometric Construction of Quiver Tensor Products

  2025cites this paper
• Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks

  2025cites this paper
• A Weight Structure on Rigid Analytic Motives over a Field

  2025cites this paper
• Generalized Clifford algebras, weighted polynomial laws, and adjunction

  2025influential citation
• The animation of the opposite of finite sets

  2025influential citation
• Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian

  2025cites this paper
• Categorical realization of collapsing subsurfaces and perverse schobers

  2025influential citation
• Cluster theory of topological Fukaya categories. Part II: Higher Teichm\"uller theory

  2025cites this paper
• Approximate Fibrations in Higher Topos Theory

  2025influential citation
• Filtrations and cohomology I: crystallization

  2025influential citation
• Every Endomorphism of the Framed Little Disk Operad is an Automorphism

  2025cites this paper
• Geometric Categories for Continuous Gauging

  2025influential citation
• A Modal Deconstruction of Löb Induction

  2025cites this paper
• Higher-dimensional generalization of abelian categories via DG-categories

  2025cites this paper
• Naturality for higher-dimensional path types

  2025cites this paper
• $\infty$-categorical group quotients via skew group algebras

  2025influential citation
• On Normed 𝔼∞-rings in Genuine Equivariant Cp-spectra

  2025cites this paper
• Combinatorics in (2,1)-categories

  2025influential citation
• Construction and Properties of the Ground State of Natural Phenomena

  2025cites this paper
• Rectification of dendroidal left fibrations

  2025influential citation
• Multiplicative structures on comodules in higher categories

  2025cites this paper
• Constructing Reedy fibrant replacements of projective fibrant simplicial presheaves

  2025influential citation
• Six-Functor Formalisms

  2025influential citation
• Fully faithful functors and pushouts of $\infty$-categories

  2025cites this paper
• Mackey functors and classical equivariant $K$-theory

  2025influential citation
• A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes

  2025cites this paper
• A Rose by Any Other Name Would Smell as Sweet: Categorical Homotopy Theory for Large Language Models

  2025cites this paper
• Existence Theorem for Cumulative Universe Towers and Its Applications

  2025cites this paper
• A Generalized Crystalline Equivalence Principle

  2025cites this paper
• Derived Stratified-Microlocal Framework and Moduli Space Resolution for the Cheeger-Goresky-Macpherson Conjecture

  2025influential citation
• On the real Section Conjecture in \'etale homotopy theory

  2025cites this paper
• Models for cyclic infinity operads

  2025influential citation
• A model for the assembly map of bordism-invariant functors

  2025influential citation
• A remark on the total simplicial set functor

  2025influential citation
• On the squares functor and the Gaitsgory-Rozenblyum conjectures

  2025influential citation
• Additivity of non-acyclicity classes for constructible \'etale sheaves

  2025cites this paper
• Recursive Difference Categories and Topos-Theoretic Universality

  2025cites this paper
• What are the extended pure inner forms of a cover?

  2025cites this paper
• Deligne 1-motives with torsion and \'etale motives

  2025cites this paper
• An illustration of formal moduli problems with differential graded Lie algebras

  2025cites this paper
• A generalization of Voevodsky’s reconstruction theorem of schemes from their étale topoi

  2025influential citation
• The Time-Energy Principle in Algebraic Geometry

  2025cites this paper
• Is Quantum Mechanics a proper subset of Classical Mechanics?

  2025cites this paper
• Topos Causal Models

  2025cites this paper
• Unital k-restricted ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-operads

  2025cites this paper
• Revisiting (∞,2)${(\infty,2)}$‐naturality of the Yoneda embedding

  2025cites this paper
• Reduced points of $\mathbb{E}_{\infty}$-rings in positive characteristic

  2025cites this paper
• Report on $\mathbb{E}_\infty$-descendability

  2025cites this paper
• $G\operatorname{-CGWH}$ spaces and their (homotopy) colimits

  2025cites this paper
• Bicategories of C*-correspondences as Dwyer-Kan localisations

  2025cites this paper
• Addendum: localization of lax symmetric monoidal categories

  2025cites this paper
• Simplicial Approach to Frobenius Algebras in the Category of Relations

  2025cites this paper
• Framed configuration spaces and exotic spheres

  2025influential citation
• Models for rational $(\infty, 1)$-categories

  2025cites this paper
• Motivic homotopy theory for perfect schemes

  2025cites this paper
• Simplicial Perspectives on (Pseudo) Effect Algebras

  2025cites this paper
• On the Mac Lane $Q$-Construction for Exact $\infty$-Categories

  2025cites this paper
• A Survey on Lawvere's Fixed-Point Theorem

  2025influential citation
• Prismatization via spherical loop spaces

  2025cites this paper
• Sketchable infinity categories

  2025influential citation
• Embedding Calculus, Goodwillie Calculus and Link Invariants

  2025cites this paper
• Quasi-canonical lifting of projective varieties in positive characteristic

  2025cites this paper