# nLab Higher Topos Theory

### Context

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

(∞,1)-category theory

## Models

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

This entry is about 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

# Summary

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

Revised on November 26, 2012 16:25:36 by Urs Schreiber (82.169.65.155)