# nLab enriched functor category

### Context

#### Enriched category theory

enriched category theory

# Contents

## 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 $\left[C,D\right]$ 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 $\left[C,D\right]$ whose

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

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

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

over the functor

$D\left(F\left(-\right),G\left(-\right)\right):{C}^{\mathrm{op}}\otimes C\to V\phantom{\rule{thinmathspace}{0ex}}.$D(F(-),G(-)) : C^{op} \otimes C \to V \,.

Write in the following ${E}_{c}:\left[C,D\right]\left(F,G\right)\to D\left(F\left(c\right),G\left(c\right)\right)$ for the canonical morphism out of the end (the counit).

• the composition operation

${\circ }_{K,F,G}:\left[C,D\right]\left(F,G\right)\otimes \left[C,D\right]\left(K,F\right)\to \left[C,D\right]\left(K,G\right)$\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 $\left[C,D\right]\left(K,F\right)$ obtained from observing that the composites

$\left[C,D\right]\left(F,G\right)\otimes \left[C,D\right]\left(K,F\right)\stackrel{{E}_{c}\otimes {E}_{d}}{\to }D\left(F\left(c\right),G\left(c\right)\right)\otimes D\left(K\left(c\right),F\left(c\right)\right)\stackrel{{\circ }_{K\left(c\right),F\left(c\right),G\left(c\right)}}{\to }D\left(K\left(c\right),F\left(c\right)\right)$[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

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

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

as an equalizer: it is the universal subobject

${\int }_{c\in C}D\left(F\left(c\right),G\left(c\right)\right)↪\prod _{c\in \mathrm{Obj}\left(C\right)}D\left(F\left(c\right),G\left(c\right)\right)$\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 $\eta \in {\int }_{c\in C}D\left(F\left(c\right),G\left(c\right)\right)$ is a collection of morphisms

$\left(F\left(c\right)\stackrel{{\eta }_{c}}{\to }G\left(c\right){\right)}_{c\in \mathrm{Obj}\left(c\right)}$( 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 $D\left(F\left(-\right),G\left(-\right)\right)$ 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 $\left(F\left(c\right)\stackrel{{\eta }_{c}}{\to }G\left(c\right)\right)↦\left(F\left(c\right)\stackrel{{\eta }_{c}}{\to }G\left(c\right)\stackrel{G\left(f\right)}{\to }G\left(d\right)\right)$

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

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

$\begin{array}{ccc}F\left(c\right)& \stackrel{{\eta }_{c}}{\to }& G\left(c\right)\\ {↓}^{F\left(f\right)}& & {↓}^{G\left(f\right)}\\ F\left(d\right)& \stackrel{{\eta }_{d}}{\to }& G\left(d\right)\end{array}$\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 ${\int }_{c\in C}D\left(F\left(c\right),G\left(c\right)\right)$ are precisely the natural transformations between $F$ and $G$.

## References

See section 2.2 p. 29 of the standard

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

Revised on March 28, 2012 05:02:40 by Urs Schreiber (82.169.65.155)