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



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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 (Rezk 98, Bergner 07).

(Another, complementary, truncation is to the homotopy 2-category of (∞,1)-categories.)


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 .


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

The nerve into simplicial spaces

The nerve functor

N:(,1)Cat 1PSh(Δ,Gpd):CnCore(C [n]) N : (\infty,1)Cat_1 \to PSh(\Delta, \infty Gpd) : C \mapsto n \mapsto Core(C^{[n]})

is fully faithful. Thus, the (,1)(\infty,1)-category of (,1)(\infty,1)-categories can be identified with the (,1)(\infty,1)-category of internal categories in Gpd \infty Gpd

This is closely related to the complete Segal space model.

NN is, in fact, the embedding of a reflective sub-(infinity,1)-category. The (,1)(\infty,1)-categories can be identified with the subcategory of PSh(Δ,Gpd)PSh(\Delta, \infty Gpd) of local objects with respect to the spine inclusions Sp nΔ nSp^n \subseteq \Delta^n and with the map J1J \to 1, where JJ is the indiscrete simplicial space on two discrete objects.

Alternatively, the map J1J \to 1 can be replaced with the projection from the simplicial discrete space formed from the union of two 2-simplices expressing the idea of a morphism with a left and right inverse fg1fg \simeq 1 and gh1gh \simeq 1.


  • 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.


In terms of complete Segal spaces:

In terms of quasi-categories:

Last revised on June 10, 2021 at 09:21:24. See the history of this page for a list of all contributions to it.