nLab
enriched functor category

Idea

In the context of $V$-enriched category theory, for $F,G:C\to D$ two $V$-enriched functors between $V$-enriched categories, the collection of natural transformations from $F$ to $G$ can also be given the structure of an object in $V$, so that the functor category, denoted $[C,D]$ in the enriched context, is itself a $V$-enriched category.

Definition

For $C$ and $D$ $V$-enriched categories, there is a $V$-enriched category denoted $[C,D]$ whose

• objects are the $V$-functors $F:C\to D$

• hom-objects $[C,D](F,G)$ between $V$-functors $F,G$ are given by the $V$-enriched end

  $$[C,D](F,G):={\int }_{c\in C}D(F(c),G(c))$$[C,D](F,G) := \int_{c \in C} D(F(c), G(c))

  over the functor

  $$D(F(-),G(-)):C^{\mathrm{op}}\otimes C\to V.$$D(F(-),G(-)) : C^{op} \otimes C \to V \,.

  Write in the following $E_c:[C,D](F,G)\to D(F(c),G(c))$ for the canonical morphism out of the end (the counit).

• the composition operation

  $${\circ }_{K,F,G}:[C,D](F,G)\otimes [C,D](K,F)\to [C,D](K,G)$$\circ_{K,F,G} : [C,D](F,G)\otimes [C,D](K,F) \to [C,D](K,G)

  is the universal morphism into the end $[C,D](K,F)$ obtained from observing that the composites

  $$[C,D](F,G)\otimes [C,D](K,F)\stackrel{E_c\otimes E_d}{\to }D(F(c),G(c))\otimes D(K(c),F(c))\stackrel{{\circ }_{K(c),F(c),G(c)}}{\to }D(K(c),F(c))$$[C,D](F,G)\otimes [C,D](K,F) \stackrel{E_c\otimes E_d}{\to} D(F(c),G(c)) \otimes D(K(c),F(c)) \stackrel{\circ_{K(c), F(c), G(c)}}{\to} D(K(c), F(c))

  form an extraordinary $V$-natural family, equivalently that

  $$[C,D](F,G)\otimes [C,D](K,F)\stackrel{\prod _{c\in \mathrm{Obj}(c)}{E}_c\otimes E_c}{\to }\prod _{c\in \mathrm{Obj}(c)}D(F(c),G(c))\otimes D(K(c),F(c))\stackrel{\prod _{c\in \mathrm{Obj}(c)}{\circ }_{K(c),F(c),G(c)}}{\to }\prod _{c\in \mathrm{Obj}(c)}D(K(c),F(c))$$[C,D](F,G)\otimes [C,D](K,F) \stackrel{\prod_{c \in Obj(c)}E_c\otimes E_c}{\to} \prod_{c \in Obj(c)} D(F(c),G(c)) \otimes D(K(c),F(c)) \stackrel{\prod_{c \in Obj(c)}\circ_{K(c), F(c), G(c)}}{\to} \prod_{c \in Obj(c)}D(K(c), F(c))

  equalizes the two maps appearing in the equalizer definition of the end.

Proposition

For $V=$Set, so that $V$-enriched categories are just ordinary locally small categories, the $V$-enriched functor category coincides with the ordinary functor category. (See the examples below.)

Examples

Ordinary functor categories

To understand the role of the end here, it is useful to spell this out for the case where $V=$ Set, where we are dealing with ordinary locally small categories.

So let $V=\mathrm{Set}$ where set is equipped with its cartesian monoidal structure.

Recall the definition of the end over

$$D(F(-),G(-)):C^{\mathrm{op}}\otimes C\to \mathrm{Set}$$D(F(-),G(-)) : C^{op} \otimes C \to Set

as an equalizer: it is the universal subobject

$${\int }_{c\in C}D(F(c),G(c))\hookrightarrow \prod _{c\in \mathrm{Obj}(C)}D(F(c),G(c))$$\int_{c \in C} D(F(c), G(c)) \hookrightarrow \prod_{c \in Obj(C)} D(F(c), G(c))

of the product of all hom-sets in $D$ between the images of objects in $C$ under the functors $F$ and $G$. So one element $\eta \in {\int }_{c\in C}D(F(c),G(c))$ is a collection of morphisms

$$(F(c)\stackrel{{\eta }_c}{\to }G(c){)}_{c\in \mathrm{Obj}(c)}$$( F(c) \stackrel{\eta_c}{\to} G(c))_{c \in Obj(c)}

such that the “left and right action” (in the sense of distributors) of $D(F(-),G(-))$ on these elements coincides. Unwrapping what this action is (see the details at end) one find that

• the “right action” by a morphism $c\stackrel{f}{\to }d$ is the postcomposition $(F(c)\stackrel{{\eta }_c}{\to }G(c))\mapsto (F(c)\stackrel{{\eta }_c}{\to }G(c)\stackrel{G(f)}{\to }G(d))$

• the “left action” by a morphism $c\stackrel{f}{\to }d$ is the precomposition $(F(d)\stackrel{{\eta }_d}{\to }G(d))\mapsto (F(c)\stackrel{F(f)}{\to }F(d)\stackrel{{\eta }_d}{\to }G(d))$.

So the invariants under the combined action are those $\eta$ for which for all $f:c\to d$ in $C$ the diagram

$$\begin{array}{ccc}F(c)& \stackrel{{\eta }_c}{\to }& G(c)\\ {\downarrow }^{F(f)}& & {\downarrow }^{G(f)}\\ F(d)& \stackrel{{\eta }_d}{\to }& G(d)\end{array}$$\array{ F(c) &\stackrel{\eta_c}{\to} & G(c) \\ \downarrow^{F(f)} && \downarrow^{G(f)} \\ F(d) &\stackrel{\eta_d}{\to} & G(d) }

commutes. Evidently, this means that the elements $\eta$ of the end ${\int }_{c\in C}D(F(c),G(c))$ are precisely the natural transformations between $F$ and $G$.

Related concepts

• enriched Reedy category, model structure on homotopical presheaves, model structure on sSet-presheaves

References

See section 2.2 p. 29 of the standard

• Max Kelly, Basic concepts of enriched category theory (pdf)