# nLab Higher Topos Theory

### Context

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

(∞,1)-category theory

## Models

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

This entry collects links related to the book

• 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.

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

# 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

# Summary

## General idea

Recall the following familiar 1-categorical statement:

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 stacks 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

## 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

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## Appendix

### A.3 Simplicial categories

#### A.3.7 Localizations of simplicial model categories

category: reference

Last revised on May 19, 2019 at 14:09:33. See the history of this page for a list of all contributions to it.