nLab
(infinity,1)-category of (infinity,1)-categories

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The collection of all (∞,1)-categories forms naturally the (∞,2)-category (∞,1)Cat.

But for many purposes it is quite sufficient to regard only invertible natural transformations between (∞,1)-functor, which means that one needs just the maximal (∞,1)-category inside that (,2)(\infty,2)-category of all (,1)(\infty,1)-categories.

Given that an (,1)(\infty,1)-category is a context for abstract homotopy theory, the (,1)(\infty,1)-category of (,1)(\infty,1)-categories is also called the the homotopy theory of homotopy theories.

Definition

Intrinsic definition

The full SSet-enriched-subcategory of SSet on those simplicial sets which are quasi-categories is – by the properties discussed at (∞,1)-category of (∞,1)-functors – itself a quasi-category-enriched category. This is the (∞,2)-category of (∞,1)-categories.

The sSet-subcategory of that obtained by picking of each hom-object the core, i.e. the maximal ∞-groupoid/Kan complex yields an ∞-groupoid/Kan complex-enriched category. This is the (,1)(\infty,1)-category of (,1)(\infty,1)-categories in its incarnation as a simplicially enriched category. Forming its homotopy coherent nerve produces the quasi-category of quasi-categories .

Models

The Joyal-model structure for quasi-categories is an sSet JoyalsSet_{Joyal}-enriched model category and hence its full SSet-subcategory on cofibrant-fibrant objects is the (,2)(\infty,2)-category of (,1)(\infty,1)-categories.

An SSet QuillenSSet_{Quillen}-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets) whose full subcategory of fibrant-cofibrant objects is the (,1)(\infty,1)-category (,1)Cat(\infty,1)Cat is the model structure on marked simplicial sets (over the terminal set). Its underlying plain model category is Quillen equivalent to the Joyal-model structure, but it is indeed sSet QuillensSet_{Quillen}-enriched.

Other model structures that present the (,1)(\infty,1)-category of all (,1)(\infty,1)-categories are

Applications

  • of particular interest is the (,1)(\infty,1)-subcategory (,1)PresCat 1(,1)Cat 1(\infty,1)PresCat_1 \hookrightarrow (\infty,1)Cat_1 of presentable (∞,1)-categories.

References

chapter 3 of

Last revised on November 26, 2018 at 03:15:45. See the history of this page for a list of all contributions to it.