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;

following an earlier sketch in

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

An online textbook of a similar content is developing at:

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 $(\infty,1)$ (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 $(\infty,1)$-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.

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