Contents
noncommutative geometry (general flavour)
Contents
n-category = (n,n)-category
n-poset = (n−1,n)-category
n-groupoid = (n,0)-category
1-categorical models for higher category theory
Contents
characterization of (∞,1)-toposes
constructions in (∞,1)-toposes
definition
constructions
refinements
presentation of
-categories
model structures
on dg-algebras/on dg-coalgebras and on on cosimplicial rings (related by monoidal Dold-Kan correspondence)
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.
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 to (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 -categorical machinery from the first part there are natural -categorical versions of -presheaf and -sheaf categories (i.e. -categories of ∞-stacks): the ”-topoi” that give the book its title (more descriptively, these would be called “Grothendieck -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 -categorical context by showing that, indeed, these are models for ∞-stack (∞,1)-toposes.
Don’t be put-off by the sheer size. On top of everything else, Lurie is a great expositor.
The book Higher topos theory together with Lurie’s work on Stable ∞-Categories is close to an -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.
-categories