(infinity,1)-category of (infinity,1)-functors



The generalization of the notion of functor category from category theory to (∞,1)-higher category theory.


Let CC and DD be (∞,1)-categories, taken in their incarnation as quasi-categories. Then

Func(C,D):=sSet(C,D) Func(C,D) := sSet(C,D)

is the simplicial set of morphisms of simplicial sets between CC and DD (in the standard sSet-enrichment of SSetSSet):

sSet(C,D):=[C,D]:=([n]Hom sSet(Δ[n]×C,D)). sSet(C,D) := [C,D] := ([n] \mapsto Hom_{sSet}(\Delta[n]\times C,D)) \,.

The objects in Fun(C,D)Fun(C,D) are the (∞,1)-functors from CC to DD, the morphisms are the corresponding natural transformations or homotopies, etc.


The simplicial set Fun(C,D)Fun(C,D) is indeed a quasi-category.

In fact, for CC and DD any simplicial sets, Fun(C,D)Fun(C,D) is a quasi-category if DD is a quasi-category.


Using that sSet is a closed monoidal category the horn filling conditions

Λ[n] i [C,D] Δ[n] \array{ \Lambda[n]_i &\to& [C,D] \\ \downarrow & \nearrow \\ \Delta[n] }

are equivalent to

C×Λ[n] i D C×Δ[n]. \array{ C \times \Lambda[n]_i &\to& D \\ \downarrow & \nearrow \\ C \times \Delta[n] } \,.

Here the vertical map is inner anodyne for inner horn inclusions Λ[n] iΔ[n]\Lambda[n]_i \hookrightarrow \Delta[n], and hence the lift exists whenever DD has all inner horn fillers, hence when DD is a quasi-category.

For the definition of (,1)(\infty,1)-functors in other models for (,1)(\infty,1)-categories see (∞,1)-functor.



The projective and injective global model structure on functors as well as the Reedy model structure if CC is a Reedy category presents (,1)(\infty,1)-categories of (,1)(\infty,1)-functors, at least when there exists a combinatorial simplicial model category model for the codomain.


Write N:sSetCatsSetN : sSet Cat \to sSet for the homotopy coherent nerve. Since this is a right adjoint it preserves products and hence we have a canonical morphism

N(C)×N([C,A])N(C×[C,A])N(ev)N(A) N(C) \times N([C,A]) \simeq N(C \times [C,A]) \stackrel{N(ev)}{\to} N(A)

induced from the hom-adjunct of Id:[C,A][C,A]Id : [C,A] \to [C,A].

The fibrant-cofibrant objects of [C,A][C,A] are enriched functors that in particular take values in fibrant cofibrant objects of AA. Therefore this restricts to a morphism

N(C)×N([C,A] )N hc(ev)N(A ). N(C) \times N([C,A]^\circ) \stackrel{N_{hc}(ev)}{\to} N(A^\circ) \,.

By the internal hom adjunction this corresponds to a morphism

N([C,A] )sSet(N hc(C),N(A )). N([C,A]^\circ) \stackrel{}{\to} sSet(N_{hc}(C), N(A^\circ)) \,.

Here A A^\circ is Kan complex enriched by the axioms of an sSet QuillensSet_{Quillen}- enriched model category, and so N(A )N(A^\circ) is a quasi-category, so that we may write this as

=Func(N(C),N(A )). \cdots = Func(N(C), N(A^\circ)) \,.

This canonical morphism

N([C,A] )Func(N(C),N(A )) N([C,A]^\circ) \stackrel{}{\to} Func(N(C), N(A^\circ))

is an (,1)(\infty,1)-equivalence in that it is a weak equivalence in the model structure for quasi-categories.

This is (Lurie, prop.


The strategy is to show that the objects on both sides are exponential objects in the homotopy category of sSet JoyalsSet_{Joyal}, hence isomorphic there.

That Func(N(C),N(A ))(N(A )) N(C)Func(N(C), N(A^\circ)) \simeq (N(A^\circ))^{N(C)} is an exponential object in the homotopy category is pretty immediate.

That the left hand is an isomorphic exponential follows from (Lurie, corollary A.3.4.12), which asserts that for CC and DD sSetsSet-enriched categories with CC cofibrant and AA as above, we have that composition with the evaluation map induces a bijection

Hom Ho(sSetCat)(D,[C,A] )Hom Ho(sSetCat)(C×D,A ). Hom_{Ho(sSet Cat)}(D, [C,A]^\circ) \stackrel{\simeq}{\to} Hom_{Ho(sSet Cat)}(C \times D, A^\circ) \,.

Since Ho(sSetCat Bergner)Ho(sSet Joyal)Ho(sSet Cat_{Bergner}) \simeq Ho(sSet_{Joyal}) this identifies also N([C,A] )N([C,A]^\circ) with the exponential object in question.

Limits and colimits

For CC an ordinary category that admits small limits and colimits, and for KK a small category, the functor category Func(D,C)Func(D,C) has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for (,1)(\infty,1)-categories of (,1)(\infty,1)-functors


Let KK and CC be quasi-categories, such that CC has all colimits indexed by KK.

Let DD be a small quasi-category. Then

  • The (,1)(\infty,1)-category Func(D,C)Func(D,C) has all KK-indexed colimits;

  • A morphism K Func(D,C)K^\triangleright \to Func(D,C) is a colimiting cocone precisely if for each object dDd \in D the induced morphism K CK^\triangleright \to C is a colimiting cocone.

This is (Lurie, corollary



A morphism α\alpha in Func(D,C)Func(D,C) (that is, a natural transformation) is an equivalence if and only if each component α d\alpha_d is an equivalence in CC.

This is due to (Joyal, Chapter 5, Theorem C).



The intrinsic definition is in section 1.2.7 of

The discussion of model category models is in A.3.4.

The theorem about equivalences is in

Revised on May 16, 2013 01:04:10 by Urs Schreiber (