nLab
(infinity,1)Cat

$(\mathrm{\infty },1)\mathrm{Cat}$ is the (∞,2)-category of all small (∞,1)-categories.

Its full subcategory on ∞-groupoids is ∞Grpd.

The $(\mathrm{\infty },2)$-category

As an $\mathrm{SSet}$-category

One incarnation of (∞,2)-categories is given by quasi-category-enriched categories (see there for details). As such $(\mathrm{\infty },1)\mathrm{Cat}$ is the full SSet-enriched subcategory of SSet on those simplicial sets that are quasi-categories. By the fact described at (∞,1)-category of (∞,1)-functors this is indeed a quasi-category-enriched category.

As an enriched model category

The model category presenting this (∞,2)-category is the Joyal model structure for quasi-categories ${\mathrm{sSet}}_{\mathrm{Joyal}}$. Its full sSet-subcategory is the quasi-category enriched category of quasi-categories from above.

The $(\mathrm{\infty },1)$-category

Sometimes it is useful to consider inside the full $(\mathrm{\infty },2)$-catgeory of $(\mathrm{\infty },1)$-categories just the maximal $(\mathrm{\infty },1)$-category and discarding all non-invertible 2-morphisms. This is the (∞,1)-category of (∞,1)-categories.

As an $\mathrm{SSet}$-category

As an SSet-enriched category the (∞,1)-category of (∞,1)-categories is obtained from the quasi-category-enriched version by picking in each hom-object simplicial set of $(\mathrm{\infty },1)\mathrm{Cat}$ the maximal Kan complex.

As an enriched model category

One model category structure presenting this is the model structure on marked simplicial sets. As a plain model category this is Quillen equivalent to ${\mathrm{sSet}}_{\mathrm{Joyal}}$, but as an enriched model category it is ${\mathrm{sSet}}_{\mathrm{Quillen}}$ enriched, so that its full SSet-subcategory on fibrant-cofibrant objects presents the $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },1)$-categories.

Properties

Limits and colimits in $(\mathrm{\infty },1)$Cat

Limits and colimits over a (∞,1)-functor with values in $(\mathrm{\infty },1)\mathrm{Cat}$ may be reformulation in terms of the universal fibration of (infinity,1)-categories $Z\to \mathrm{\infty }{\mathrm{Grpd}}^{\mathrm{op}}$

Then let $X$ be any (∞,1)-category and

$$F:X\to (\mathrm{\infty },1)\mathrm{Cat}$$F : X \to (\infty,1)Cat

an (∞,1)-functor. Recall that the coCartesian fibration $E_F\to X$ classified by $F$ is the pullback of the universal fibration of (∞,1)-categories $Z$ along F:

$$\begin{array}{ccc}{E}_F& \to & Z\\ \downarrow & & \downarrow \\ X& \stackrel{F}{\to }& (\mathrm{\infty },1)\mathrm{Cat}\end{array}$$\array{ E_F &\to& Z \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& (\infty,1)Cat }

Proposition

Let the assumptions be as above. Then:

• The colimit of $F$ is equivalent to $E_F$:

  $$E_F\simeq \mathrm{colim}F$$E_F \simeq colim F

• The limit of $F$ is equivalent to the (infinity,1)-category of cartesian section of $E_F\to X$

  $${\Gamma }_X(E_F)\simeq \lim F.$$\Gamma_X(E_F) \simeq lim F \,.

Proof

This is HTT, section 3.3.

Automorphisms

Theorem

The full subcategory of the (∞,1)-category of (∞,1)-categories $\mathrm{Func}((\mathrm{\infty },1)\mathrm{Cat},(\mathrm{\infty },1)\mathrm{Cat})$ on those (∞,1)-functors that are equivalences is equivalent to $\{\mathrm{Id},\mathrm{op}\}$: it contains only the identity functor and the one that sends and $(\mathrm{\infty },1)$-category to its opposite (infinity,1)-category.

Proof

This is due to

• Bertrand Toen, Vers une axiomatisation de la théorie des catégories supérieures , K-theory 34 (2005), no. 3, 233-263.

It appears as HTT, theorem 5.2.9.1 (arxiv v4+ only)

First of all the statement is true for the ordinary category of posets. This is prop. 5.2.9.14.

From this the statement is deduced for $(\mathrm{\infty },1)$ -categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, prop. 5.2.9.11, which means that posets $C$ are characterized by the fact that

$${\pi }_0(\mathrm{\infty },1)\mathrm{Cat}(D,C)\to {\mathrm{Hom}}_{\mathrm{Set}}({\pi }_0(\mathrm{\infty },1)\mathrm{Cat}(*,D),{\pi }_0(\mathrm{\infty },1)\mathrm{Cat}(*,C))$$\pi_0 (\infty,1)Cat(D,C) \to Hom_{Set}( \pi_0 (\infty,1)Cat(*,D) , \pi_0 (\infty,1)Cat(*,C) )

is an injection for all $D\in (\mathrm{\infty },1)\mathrm{Cat}$.

This is preserved under automorphisms of $(\mathrm{\infty },1)\mathrm{Cat}$, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or $(-{)}^{\mathrm{op}}$ there, by the above statement for posets.

Now finally the main point of the proof is to see that the linear posets $\Delta \subset (\mathrm{\infty },1)\mathrm{Cat}$ are dense in $(\mathrm{\infty },1)\mathrm{Cat}$, i.e. that the identity transformation of the inclusion functor $\Delta \hookrightarrow (\mathrm{\infty },1)\mathrm{Cat}$ exhibits ${\mathrm{Id}}_{(\mathrm{\infty },1)\mathrm{Cat}}$ as the left Kan extension

$$\begin{array}{ccc}\Delta & \hookrightarrow & (\mathrm{\infty },1)\mathrm{Cat}\\ \downarrow & {\nearrow }_{\mathrm{Lan}=\mathrm{Id}}\\ (\mathrm{\infty },1)\mathrm{Cat}\end{array}.$$\array{ \Delta &\hookrightarrow& (\infty,1)Cat \\ \downarrow & \nearrow_{Lan = \mathrlap{Id}} \\ (\infty,1)Cat } \,.

Presentations

The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $𝒪$-monoidal (∞,1)-categories (fourth table).

	general pattern
strict enrichment		(∞,1)-category/(∞,1)-operad
$\downarrow$		$\downarrow$
enriched (∞,1)-category	$\hookrightarrow$	internal (∞,1)-category
	(∞,1)Cat
SimplicialCategories	$-$homotopy coherent nerve$\to$	SimplicialSets/quasi-categories		RelativeSimplicialSets
$\downarrow$simplicial nerve		$\downarrow$
SegalCategories	$\hookrightarrow$	CompleteSegalSpaces
	(∞,1)Operad
SimplicialOperads	$-$homotopy coherent dendroidal nerve$\to$	DendroidalSets		RelativeDendroidalSets
$\downarrow$dendroidal nerve		$\downarrow$
SegalOperads	$\hookrightarrow$	DendroidalCompleteSegalSpaces
	$𝒪$Mon(∞,1)Cat
		DendroidalCartesianFibrations

Related concepts

• Cat, Operad

• 2Cat

• $(\mathrm{\infty },1)$Cat, (∞,1)Operad

• (∞,n)Cat

category: category