Higher Topos Theory


(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

This entry collects links related to the book

which discusses the higher category theory of (∞,1)-categories in general and that of (∞,1)-categories of (∞,1)-sheaves (i.e. of ∞-stacks) – called (Grothendieck-Rezk-Lurie) (∞,1)-toposes – in particular.

One should beware that the arXiv version of this book has been updated since the publication of the print version, including addition of some new material!


Related entries

For general information on higher category and higher topos theory see also

If you need basics, see

If you need more motivation see

If you need to see applications see for instance


General idea

Recall the following familiar 1-categorical statement:

One can think of Lurie’s book as a comprehensive study of the generalization of the above statement from 11 to (,1)(\infty,1) (recall the notion of (n,r)-category):

First part, sections 1-4

Based on work by André Joyal on the quasi-category model for (∞,1)-categories, Lurie presents a comprehensive account of the theory of (∞,1)-categories including the definitions and properties of all the standard items familiar from category theory (limits, fibrations, etc.)

Second part, sections 5-7

Given the (,1)(\infty,1)-categorical machinery from the first part there are natural (,1)(\infty,1)-categorical versions of (,1)(\infty,1)-presheaf and (,1)(\infty,1)-sheaf categories (i.e. (,1)(\infty,1)-categories of ∞-stacks): the ”\infty-topoi” that give the book its title (more descriptively, these would be called “Grothendieck (,1)(\infty,1)-topoi”). Lurie investigates their properties in great detail and thereby in particular puts the work by Brown, Joyal, Jardine, Toën on the model structure on simplicial presheaves into a coherent (,1)(\infty,1)-categorical context by showing that, indeed, these are models for ∞-stack (∞,1)-toposes.

How to read the book

1-categorical background

The book Higher topos theory together with Lurie’s work on Stable ∞-Categories is close to an (,1)(\infty,1)-categorical analog of the 1-categorical material as presented for instance in

Sections with crucial concepts

The book discusses crucial concepts and works out plenty of detailed properties. On first reading it may be helpful to skip over all the technical parts and pick out just the central conceptual ideas. These are the following:


1 An overview of higher category theory

2 Fibrations of Simplicial Sets

2.1 Left fibrations

2.2 Simplicial categories and \infty-categories

2.3 Inner fibrations

2.3.1 Correspondences

2.3.2 Stability properties of inner fibrations


2.3.3 Minimal fibrations

2.3.4 nn-Categories

2.4 Cartesian fibrations

2.4.1 Cartesian morphisms

2.4.2 Cartesian fibrations

2.4.3 Stability properties of Cartesian fibrations

2.4.4 Mapping spaces and Cartesian fibrations

2.4.5 Application: Invariance of Undercategories

2.4.6 Application: Categorical fibrations over a point

2.4.7 Bifibrations

3 The \infty-Category of \infty-Categories

4 Limits and Colimits

4.1 Cofinality

4.2 Techniques for computing colimits

4.3 Kan extensions

4.3.1 Relative colimits

4.4 Examples of colimits

5 Presentable and Accessible \infty-Categories

5.1 (,1)(\infty,1)-Categories of presheaves

5.2 Adjoint (,1)(\infty,1)-functors

5.2.8 Factorization systems

5.3 (,1)(\infty,1)-Categories of inductive limits

5.3.1 Filtered \infty-categories

5.3.2 Right exactness

5.3.3 Filtered colimits

5.3.4 Compact objects

5.3.5 Ind-objects

5.3.6 Adjoining colimits to \infty-categories

5.4 Accessible (,1)(\infty,1)-categories

5.4.1 Locally small \infty-categories

5.4.2 Accessible (,1)(\infty,1)-categories

5.4.3 Accessible and idempotent-complete (,1)(\infty,1)-categories

5.5 Presentable (,1)(\infty,1)-categories

5.5.1 Presentability

5.5.2 Representable functors and the adjoint functor theorem

5.5.3 Limits and colimits of presentable \infty-categories

5.5.4 Local objects

5.5.5 Factorization systems on presentable \infty-categories

5.5.6 Truncated objects

5.5.7 Compactly generated \infty-categories

5.5.8 Nonabelian Derived Categories

5.5.9 Quillen’s model for 𝒫 Σ(C)\mathcal{P}_\Sigma(C)

6 \infty-Topoi

6.1 Definitions and characterizations

6.1.1 Giraud’s Axioms in the \infty-Categorical setting

6.1.2 Groupoid objects

6.1.3 \infty-Topoi and descent

6.1.4 Free Groupoids


6.1.5 Giraud’s theorem for \infty-Topoi

6.1.6 \infty-Topoi and classifying objects

6.2 Constructions of (,1)(\infty,1)-toposes

6.2.1 Left exact localization

6.2.2 Grothendieck topologies and sheaves in higher category theory

6.2.3 Effective epimorphisms

6.3 The \infty-Category of \infty-Topoi

6.3.1 Geometric morphisms

6.3.2 Colimits of \infty-topoi

6.3.3 Filtered limits of \infty-topoi

6.3.4 General limits of \infty-topoi

6.3.5 Etale Morphisms of \infty-topoi

6.4 nn-Topoi

6.5 Homotopy theory in an (,1)(\infty,1)-topos

6.5.1 Homotopy groups

6.5.2 \infty-Connectedness

6.5.3 Hypercovering

6.5.4 Descent versus Hyperdescent

7 Higher Topos Theory in Topology

7.1 Paracompact spaces

7.2 Dimension theory


A.1 Category theory

A.2 Model categories

A.3 Simplicial categories

A.3.1 Enriched and monoidal model categoires

A.3.2 The model structure on S\mathbf{S}-enriched categories

A.3.3 Model structures on diagram categories

A.3.4 Path spaces in S\mathbf{S}-enriched categories

A.3.5 Homotopy colimits of S\mathbf{S}-enriched categories

A.3.5 Exponentiation in model categories

A.3.7 Localizations of simplicial model categories

category: reference

Revised on February 15, 2014 05:05:43 by Urs Schreiber (