nLab (infinity,1)Cat

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Models

categories of categories

$(n+1,r+1)$-categories of (n,r)-categories

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

Its full subcategory on ∞-groupoids is ∞Grpd.

Contents

The $(\infty,2)$-category

As an $SSet$-category

One incarnation of (∞,2)-categories is given by quasi-category-enriched categories (see there for details). As such $(\infty,1)Cat$ 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 $sSet_{Joyal}$. Its full sSet-subcategory is the quasi-category enriched category of quasi-categories from above.

The $(\infty,1)$-category

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

As an $SSet$-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 $(\infty,1)Cat$ 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 $sSet_{Joyal}$, but as an enriched model category it is $sSet_{Quillen}$ enriched, so that its full SSet-subcategory on fibrant-cofibrant objects presents the $(\infty,1)$-category of $(\infty,1)$-categories.

Properties

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

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

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

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

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:

$\array{ E_F &\to& Z \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& (\infty,1)Cat }$
Proposition

Let the assumptions be as above. Then:

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

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

$\Gamma_X(E_F) \simeq lim F \,.$
Proof

This is HTT, section 3.3.

Automorphisms

Theorem

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

$\pi_0 (\infty,1)Cat(D,C) \to Hom_{Set}( \pi_0 (\infty,1)Cat(*,D) , \pi_0 (\infty,1)Cat(*,C) )$

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

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

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

$\array{ \Delta &\hookrightarrow& (\infty,1)Cat \\ \downarrow & \nearrow_{Lan = \mathrlap{Id}} \\ (\infty,1)Cat } \,.$

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 $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).

general pattern
$\downarrow$$\downarrow$
enriched (∞,1)-category$\hookrightarrow$internal (∞,1)-category
(∞,1)Cat
SimplicialCategories$-$homotopy coherent nerve$\to$SimplicialSets/quasi-categoriesRelativeSimplicialSets
$\downarrow$simplicial nerve$\downarrow$
SegalCategories$\hookrightarrow$CompleteSegalSpaces
SimplicialOperads$-$homotopy coherent dendroidal nerve$\to$DendroidalSetsRelativeDendroidalSets
$\downarrow$dendroidal nerve$\downarrow$
SegalOperads$\hookrightarrow$DendroidalCompleteSegalSpaces
$\mathcal{O}$Mon(∞,1)Cat