Higher Topos Theory in nLab
Context
(∞,1)(\infty,1)-Category theory
Background
Basic concepts
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equivalences in/of (∞,1)(\infty,1)-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
(∞,1)(\infty,1)-Topos Theory
This entry collects links related to the book
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Higher Topos Theory
Annals of Mathematics Studies 170
Princeton University Press (2009)
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:
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PDF of published version from Lurie’s web site
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arXiv:math.CT/0608040 – this has been updated since the publication of the print version, including addition of some new material!
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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:
Contents
- 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 nn-Categories
- 2.4 Cartesian fibrations
- 3 The ∞\infty-Category of ∞\infty-Categories
- 4 Limits and Colimits
- 5 Presentable and Accessible ∞\infty-Categories
- 5.1 (∞,1)(\infty,1)-Categories of presheaves
- 5.2 Adjoint (∞,1)(\infty,1)-functors
- 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
- 7 Higher Topos Theory in Topology
- 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 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
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
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 11 to (∞,1)(\infty,1) (recall the notion of (n,r)-category):
- Working in the (∞,1)(\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 (∞,1)(\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 (∞,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
- 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:
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section 1.1 : the concept of (∞,1)-category
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section 5.1: the concept of (∞,1)-presheaves
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section 6.1: the concept of (∞,1)-topoi
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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
-
-
constructions in quasi-categories
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
Appendix
A.1 Category theory
A.2 Model categories
-
-
-
localization of a simplicial model category?
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
-
localization of a simplicial model category?
Last revised on February 12, 2025 at 22:47:07. See the history of this page for a list of all contributions to it.