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

Models

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 a combinatorial simplicial model category model for the codomain.

Let

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

• $A$ a combinatorial simplicial model category and $A^\circ$ its full sSet-subcategory of fibrant cofibrant objects;

• $[C,A]$ the sSet-enriched functor category equipped with either the injective or projective global model structure on functors – here: the injective or injective model structure on sSet-enriched presheaves – and $[C,A]^\circ$ its full sSet-subcategory on fibrant-cofibrant objects.

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

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

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

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

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

By the internal hom adjunction this corresponds to a morphism

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

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, so that we may write this as

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

Proposition

This canonical morphism

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

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

This is (Lurie, prop. 4.2.4.4).

Proof

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

That $Func(N(C), N(A^\circ)) \simeq (N(A^\circ))^{N(C)}$ 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 $C$ and $D$ $sSet$-enriched categories with $C$ cofibrant and $A$ as above, we have that composition with the evaluation map induces a bijection

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

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

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

Propositon

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

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, Chapter 5, Theorem C).

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 is in

• André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf)