Context
$(\infty,1)$Category theory
(∞,1)category theory
Background
Basic concepts
Universal constructions
Local presentation
Theorems
Models
$(\infty,1)$Topos Theory
(∞,1)topos theory
Background
Definitions

elementary (∞,1)topos

(∞,1)site

reflective sub(∞,1)category

(∞,1)category of (∞,1)sheaves

(∞,1)topos

(n,1)topos, ntopos

(∞,1)quasitopos

(∞,2)topos

(∞,n)topos
Characterization
Morphisms
Extra stuff, structure and property

hypercomplete (∞,1)topos

over(∞,1)topos

nlocalic (∞,1)topos

locally nconnected (n,1)topos

structured (∞,1)topos

locally ∞connected (∞,1)topos, ∞connected (∞,1)topos

local (∞,1)topos

cohesive (∞,1)topos
Models
Constructions
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 (GrothendieckRezkLurie) (∞,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!
Contents
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 1categorical statement:
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):
First part, sections 14
Based on work by André Joyal on the quasicategory 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 57
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
1categorical background
The book Higher topos theory together with Lurie’s work on Stable ∞Categories is close to an $(\infty,1)$categorical analog of the 1categorical 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:
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
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 Indobjects
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 idempotentcomplete $(\infty,1)$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