nLab Higher Topos Theory

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Context

$(\infty,1)$ -Category theory

$(\infty,1)$ -Topos Theory

This entry collects links related to the book

Jacob Lurie:

Higher Topos Theory

Annals of Mathematics Studies 170

Princeton University Press (2009)

pup:8957,

pdf

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;

following an earlier sketch in

Jacob Lurie, On $\infty$ -Topoi (arXiv:math/0306109)

The book is available online from the arXiv and also from Lurie’s web site:

PDF of published version from Lurie’s web site
arXiv:math.CT/0608040 – this has been updated since the publication of the print version, including addition of some new material!
updated version from Lurie’s web site – more recent even than the arXiv version, as of 2019

An online textbook of a similar content is developing at:

Jacob Lurie, Kerodon

General idea
First part, sections 1-4
Second part, sections 5-7
1-categorical background
Sections with crucial concepts
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 $n$ -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 $(\infty,1)$ -Categories of presheaves
5.2 Adjoint $(\infty,1)$ -functors

5.2.8 Factorization systems

5.3 $(\infty,1)$ -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 $(\infty,1)$ -categories

5.4.1 Locally small $\infty$ -categories
5.4.2 Accessible $(\infty,1)$ -categories
5.4.3 Accessible and idempotent-complete $(\infty,1)$ -categories

5.5 Presentable $(\infty,1)$ -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 $\mathcal{P}_\Sigma(C)$

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 $(\infty,1)$ -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 $n$ -Topoi
6.5 Homotopy theory in an $(\infty,1)$ -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

Appendix

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 $\mathbf{S}$ -enriched categories
A.3.3 Model structures on diagram categories
A.3.4 Path spaces in $\mathbf{S}$ -enriched categories
A.3.5 Homotopy colimits of $\mathbf{S}$ -enriched categories
A.3.5 Exponentiation in model categories
A.3.7 Localizations of simplicial model categories

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

motivation for sheaves, cohomology and higher stacks.

If you need to see applications see for instance

Summary

General idea

Recall the following familiar 1-categorical statement:

Working in the 1-category Set of 0-categories is the same as doing set theory. The point of categories and sheaves is to pass to parameterized 0-categories, namely presheaf categories: these topoi behave much like the category Set but their objects are generalized spaces that may carry more structure, for instance they may be generalized smooth spaces if one considers (pre)sheaves on Diff.

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

Working in the $(\infty,1)$ -category ∞Grpd of (∞,0)-categories is the same as doing topology. The point of ∞-stacks is to pass to parameterized (∞,0)-categories, namely (∞,1)-presheaf categories: these (∞,1)-topoi behave much like the $(\infty,1)$ category ∞Grpd but their objects are generalized spaces with higher homotopies that may carry more structure, for instance they may be $\infty$ - differentiable stack if one considers ∞-stacks on Diff.

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 $(\infty,1)$ -categorical machinery from the first part there are natural $(\infty,1)$ -categorical versions of $(\infty,1)$ -presheaf and $(\infty,1)$ -sheaf categories (i.e. $(\infty,1)$ -categories of ∞-stacks): the “ $\infty$ -topoi” that give the book its title (more descriptively, these would be called “Grothendieck $(\infty,1)$ -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 $(\infty,1)$ -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 $(\infty,1)$ -categorical analog of the 1-categorical material as presented for instance in

Kashiwara and Shapira, Categories and Sheaves.

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:

section 1.1 : the concept of (∞,1)-category
section 5.1: the concept of (∞,1)-presheaves
section 6.1: the concept of (∞,1)-topoi
section 6.2 section 6.5 and : relation to the Brown-Joyal-Jardine-Toën theory of models for ∞-stack (∞,1)-toposes in terms of the model structure on simplicial presheaves.

Content

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

inner fibration

2.3.1 Correspondences

2.3.2 Stability properties of inner fibrations

(…)

2.3.3 Minimal fibrations

minimal inner fibration

2.3.4 $n$ -Categories

2.4 Cartesian fibrations

2.4.1 Cartesian morphisms

Cartesian morphism

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

codomain fibration

3 The $\infty$ -Category of $\infty$ -Categories

4 Limits and Colimits

4.1 Cofinality

cofinal (∞,1)-functor

4.2 Techniques for computing colimits

homotopy limit

4.3 Kan extensions

4.3.1 Relative colimits

relative (infinity,1)-colimit

…

4.4 Examples of colimits

…

5 Presentable and Accessible $\infty$ -Categories

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

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

adjoint (∞,1)-functor

5.2.8 Factorization systems

orthogonal factorization system in an (∞,1)-category

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

5.3.1 Filtered $\infty$ -categories

5.3.2 Right exactness

exact (∞,1)-functor

5.3.3 Filtered colimits

5.3.4 Compact objects

compact object in an (∞,1)-category

5.3.5 Ind-objects

5.3.6 Adjoining colimits to $\infty$ -categories

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

5.4.1 Locally small $\infty$ -categories

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

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

idempotent-complete (∞,1)-category

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

5.5.5 Factorization systems on presentable $\infty$ -categories

5.5.6 Truncated objects

5.5.7 Compactly generated $\infty$ -categories

compactly generated (∞,1)-category

5.5.8 Nonabelian Derived Categories

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

model structure for (∞,1)-algebras over (∞,1,)-algebraic theories

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

Giraud's theorem

6.1.6 $\infty$ -Topoi and classifying objects

object classifier in an (infinity,1)-topos

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

étale geometric morphism

6.4 $n$ -Topoi

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

Appendix

A.1 Category theory

A.2 Model categories

homotopy theory
category with weak equivalences
homotopical category
- enriched homotopical category
- category of fibrant objects
- Waldhausen category
- simplicial localization
- model category
  - Quillen adjunction
    - Quillen equivalence
  - Quillen bifunctor
  - monoidal model category
  - enriched model category
  - localization of a simplicial model category?
    - Bousfield localization
- Reedy model structure

A.3 Simplicial categories

presentable (∞,1)-category

A.3.1 Enriched and monoidal model categoires

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

A.3.3 Model structures on diagram categories

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

(∞,1)-category of (∞,1)-functors

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

…

A.3.5 Exponentiation in model categories

…

A.3.7 Localizations of simplicial model categories

Bousfield localization of model categories
localization of a simplicial model category?
Combinatorial model categories have presentations

category: reference

Last revised on March 14, 2024 at 04:27:08. See the history of this page for a list of all contributions to it.

nLab Higher Topos Theory

Context

(∞,1)(\infty,1)-Category theory

(∞,1)(\infty,1)-Topos Theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

Contents