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

$$\mathrm{Func}(C,D):=\mathrm{sSet}(C,D)$$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 $\mathrm{SSet}$):

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

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

Proposition

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

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

Proof

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

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

are equivalent to

$$\begin{array}{ccc}C\times \Lambda [n{]}_i& \to & D\\ \downarrow & \nearrow \\ C\times \Delta [n]\end{array}.$$\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 $(\mathrm{\infty },1)$-functors in other models for $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-categories of $(\mathrm{\infty },1)$-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 $N:\mathrm{sSet}\mathrm{Cat}\to \mathrm{sSet}$ for the homotopy coherent nerve. Since this is a right adjoint it preserves products and hence we have a canonical morphism

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

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

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

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

$$N([C,A{]}^{\circ })\stackrel{}{\to }\mathrm{sSet}(N_{\mathrm{hc}}(C),N(A^{\circ })).$$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 ${\mathrm{sSet}}_{\mathrm{Quillen}}$- enriched model category, and so $N(A^{\circ })$ is a quasi-category, so that we may write this as

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

Proposition

This canonical morphism

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

is an $(\mathrm{\infty },1)$-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 ${\mathrm{sSet}}_{\mathrm{Joyal}}$, hence isomorphic there.

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

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

Since $\mathrm{Ho}(\mathrm{sSet}{\mathrm{Cat}}_{\mathrm{Bergner}})\simeq \mathrm{Ho}({\mathrm{sSet}}_{\mathrm{Joyal}})$ 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 $\mathrm{Func}(D,C)$ has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for $(\mathrm{\infty },1)$-categories of $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-category $\mathrm{Func}(D,C)$ has all $K$-indexed colimits;

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

This is (Lurie, corollary 5.1.2.3).

Examples

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

  $$({\mathrm{sSet}}_{\mathrm{Quillen}}{)}^{\circ }\simeq \mathrm{\infty }\mathrm{Grpd}$$(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 $C^{\mathrm{op}}$ presents the (∞,1)-category of (∞,1)-presheaves on $C$:

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

Reference

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.