equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
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
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:
PDF of published version from Lurie’s web site
arXiv:math.CT/0608040 – this has been updated since the publication of the print version, including addition of some new material!
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:
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
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):
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.)
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.
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
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:
section 1.1 : the concept of (∞,1)-category
section 5.1: the concept of (∞,1)-presheaves
section 6.1: the concept of (∞,1)-topoi
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.
constructions in quasi-categories
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Last revised on July 14, 2021 at 10:07:51. See the history of this page for a list of all contributions to it.