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

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 $(\infty,2)$-category of all $(\infty,1)$-categories.

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

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 $(\infty,1)$-category of $(\infty,1)$-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 $sSet_{Joyal}$-enriched model category and hence its full SSet-subcategory on cofibrant-fibrant objects is the $(\infty,2)$-category of $(\infty,1)$-categories.

An $SSet_{Quillen}$-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets) whose full subcategory of fibrant-cofibrant objects is the $(\infty,1)$-category $(\infty,1)Cat$ 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 $sSet_{Quillen}$-enriched.

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

• model structure on categories with weak equivalences

• model structure on simplicially enriched categories

• model structure for complete Segal spaces.

The nerve into simplicial spaces

The nerve functor

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

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

This is closely related to the complete Segal space model.

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

Alternatively, the map $J \to 1$ 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 $fg \simeq 1$ and $gh \simeq 1$.

Applications

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

Related concepts

• homotopy 2-category of (∞,1)-categories

References

In terms of complete Segal spaces:

• Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), 973-1007 (arXiv:math/9811037, doi:10.1090/S0002-9947-00-02653-2)

• Julia Bergner, Three models for the homotopy theory of homotopy theories, Topology Volume 46, Issue 4, September 2007, Pages 397-436 (arXiv:math/0504334, doi:10.1016/j.top.2007.03.002)

In terms of quasi-categories:

• Jacob Lurie, Chapter 3 of: Higher Topos Theory (2009)