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

Its full subcategory on ∞-groupoids is ∞Grpd.


The (,2)(\infty,2)-category

As an SSetSSet-category

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

The (,1)(\infty,1)-category

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

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


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

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

Then let XX be any (∞,1)-category and

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

an (∞,1)-functor. Recall that the coCartesian fibration E FXE_F \to X classified by FF is the pullback of the universal fibration of (∞,1)-categories ZZ along F:

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

Let the assumptions be as above. Then:

  • The colimit of FF is equivalent to E FE_F:

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

    Γ X(E F)limF. \Gamma_X(E_F) \simeq lim F \,.

This is HTT, section 3.3.



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


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 (arxiv v4+ only)

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

From this the statement is deduced for (,1)(\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., which means that posets CC are characterized by the fact that

π 0(,1)Cat(D,C)Hom Set(π 0(,1)Cat(*,D),π 0(,1)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(,1)CatD \in (\infty,1)Cat.

This is preserved under automorphisms of (,1)Cat(\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(-)^{op} there, by the above statement for posets.

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

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


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

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
enriched (∞,1)-category\hookrightarrowinternal (∞,1)-category
SimplicialCategories-homotopy coherent nerve\toSimplicialSets/quasi-categoriesRelativeSimplicialSets
\downarrowsimplicial nerve\downarrow
SimplicialOperads-homotopy coherent dendroidal nerve\toDendroidalSetsRelativeDendroidalSets
\downarrowdendroidal nerve\downarrow

category: category

Last revised on October 27, 2017 at 07:05:49. See the history of this page for a list of all contributions to it.