nLab (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.


Model category presentation

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 combinatorial simplicial-structure model on the codomain model category.


SinceN\;N is a right adjoint it preserves products so that we obtain a morphism

N(C)×N([C,A])N(C×[C,A])N(ev)N(A) N(C) \times N\big([C,A]\big) \xrightarrow{\; \sim \;} N\big(C \times [C,A]\big) \xrightarrow{\; N(ev) \;} N(A)

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

Noticing that the bifibrant objects of [C,A][C,A] are enriched functors that, in particular, take values in bifibrant objects of AA, this restricts to a morphism of the form

N(C)×N([C,A] )N(ev)N(A ), N(C) \times N\big([C,A]^\circ\big) \xrightarrow{\;\; N(ev) \;\;} N\big(A^\circ\big) \,,

which, by the internal hom-adjunction, corresponds to a morphism

N([C,A] )sSet(N(C),N(A )). N\big([C,A]^\circ\big) \xrightarrow{\phantom{--}} sSet\big( N(C) ,\, N(A^\circ) \big) \,.

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

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


This canonical morphism

(1)N([C,A] )Func(N(C),N(A )) N\big([C,A]^\circ\big) \xrightarrow{\phantom{---}} Func\big( N(C) ,\, N(A^\circ) \big)

is an equivalence of \infty -categories in that it is a weak equivalence in the model structure for quasi-categories.

This is Lurie (2009), Prop.


The strategy is to show that the objects on both sides are both exponential objects in the homotopy category of sSet Joyal sSet_{Joyal} , which, by the uniqueness of adjoints, implies that they are isomorphic in the homotopy category, which finally is equivalent to the statement to be proven.

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

That the left hand is an isomorphic exponential follows from Lurie 09, corollary A.3.4.12, which asserts that for CC and DD sSet-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)}\big(D, [C,A]^\circ\big) \xrightarrow{\simeq} Hom_{Ho(sSet Cat)}\big(C \times D, A^\circ\big) \,.

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

The following proof is fresh, still needs double-checking.


In the above situation, consider an sSet-enriched functor f:CCf \,\colon\, C' \longrightarrow C between any small sSet-enriched categories. Then under the identification of Prop. the two \infty-functors given by

  1. homotopy coherent nerve of the derived functor f *\mathbb{R}f^\ast of precomposition with ff

  2. the precomposition with the homotopy coherent nerve of ff

are related by a natural equivalence of \infty-functors:


Consider the following diagram of simplicial sets (the outer ones being quasi-categories):

Here QQ denotes any functorial cofibrant replacement (which exists, by this Example, since [C,A][C',A] is a combinatorial model category by the above discussion) and the double arrow denotes (the image under the hc-nerve of) the natural transformation with components the corresponding resolution equivalences Q(-)(-)Q(\text{-}) \xrightarrow{\;\sim\;} (\text{-}) (which are components of a natural transformation, by the nature of functorial factorization).

The left square commutes by the construction of right derived functors of right Quillen functors (eg. this Prop.) and the middle square is the naturality square of the comparison map discussed above. The remaining outer squares just exhibit the restriction to bifibrant objects, as discussed above.

The total diagram is of the claimed from. It just remains to see that the 2-morphism filling it is really an equivalence, but this follows by Prop. .

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



(equivalences of \infty-functors detected on objects)
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 (2008), Chapter 5, Theorem C (p. 125).



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 (Prop. ) is due to:

  1. Lurie’s Remark claims that Prop. also works also for the injective structure, but this is not so clear, see also MO:q/440965.

Last revised on May 15, 2023 at 07:37:37. See the history of this page for a list of all contributions to it.