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

Idea

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

Definition

Let $C$ and $D$ be (∞,1)-categories, taken in their incarnation as quasi-categories. Then

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

is the simplicial set of morphisms of simplicial sets between $C$ and $D$ (in the standard sSet-enrichment of $SSet$):

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

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

Proposition

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

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

Proof

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

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

are equivalent to

$$\array{ C \times \Lambda[n]_i &\to& D \\ \downarrow & \nearrow \\ C \times \Delta[n] } \,.$$

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

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

Properties

Model category presentation

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

Let

• $C$ be a small sSet-enriched category;

• $A$ a combinatorial simplicial model category and

  $A^\circ$ its full sSet-subcategory of bifibrant objects;

• $[C,A]$ the sSet-enriched functor category equipped with the projective¹ global model structure on functors, and

  $[C,A]^\circ$ its full sSet-subcategory on bifibrant objects.

• $N \colon sSet\text{-}Cat \to sSet$ the homotopy coherent nerve.

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

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

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

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

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

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

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

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

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

Proposition

This canonical morphism

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

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. 4.2.4.4.

Proof

The strategy is to show that the objects on both sides are both exponential objects in the homotopy category of $sSet_{Joyal}$, 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 $Func\big(N(C), N(A^\circ)\big) \simeq \big(N(A^\circ)\big)^{N(C)}$ 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 $C$ and $D$ sSet-enriched categories with $C$ cofibrant and $A$ as above, we have that composition with the evaluation map induces a bijection

$$Hom_{Ho(sSet Cat)}\big(D, [C,A]^\circ\big) \xrightarrow{\simeq} Hom_{Ho(sSet Cat)}\big(C \times D, A^\circ\big) \,.$$

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

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

Corollary

In the above situation, consider an sSet-enriched functor $f \,\colon\, C' \longrightarrow C$ 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 $\mathbb{R}f^\ast$ of precomposition with $f$

2. the precomposition with the homotopy coherent nerve of $f$

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

Proof

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

Here $Q$ denotes any functorial cofibrant replacement (which exists, by this Example, since $[C',A]$ 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 $Q(\text{-}) \xrightarrow{\;\sim\;} (\text{-})$ (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 $C$ an ordinary category that admits small limits and colimits, and for $K$ a small category, the functor category $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for $(\infty,1)$-categories of $(\infty,1)$-functors

Proposition

Let $K$ and $C$ be quasi-categories, such that $C$ has all colimits indexed by $K$.

Let $D$ be a small quasi-category. Then

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

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

This is (Lurie, corollary 5.1.2.3).

Equivalences

Proposition

(equivalences of $\infty$-functors detected on objects)
A morphism $\alpha$ in $Func(D,C)$ (that is, a natural transformation) is an equivalence if and only if each component $\alpha_d$ is an equivalence in $C$.

This is due to Joyal (2008), Chapter 5, Theorem C (p. 125).

Examples

• Between ordinary categories, it reproduces the ordinary category of functors.

• Since the standard model structure on simplicial sets presents ∞Grpd

  $$(sSet_{Quillen})^\circ \simeq \infty Grpd$$

  the model structure on simplicial presheaves (more precisely and more generally the model structure on sSet-enriched presheaves) on the opposite (∞,1)-category $C^{op}$ presents the (∞,1)-category of (∞,1)-presheaves on $C$:

  $$N([C^{op},sSet_{Quillen}]^\circ) \simeq Func(C^{op},\infty Grpd) = PSh_{(\infty,1)}(C) \,.$$

Related concepts

• functor category

• 2-functor 2-category

References

The intrinsic definition is in section 1.2.7 of

• Jacob Lurie, Higher Topos Theory

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

The theorem about equivalences (Prop. ) is due to:

• André Joyal, The Theory of Quasi-Categories and its Applications, lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM (2008) [pdf, pdf]

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1. Lurie’s Remark 4.2.4.5 claims that Prop. also works also for the injective structure, but this is not so clear, see also MO:q/440965. ↩