# 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 $[C,D]$ 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

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

over the functor

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

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

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

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

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

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

$[C,D](F,G)\otimes [C,D](K,F) \stackrel{\prod_{c \in Obj(c)}E_c\otimes E_c}{\to} \prod_{c \in Obj(c)} D(F(c),G(c)) \otimes D(K(c),F(c)) \stackrel{\prod_{c \in Obj(c)}\circ_{K(c), F(c), G(c)}}{\to} \prod_{c \in Obj(c)}D(K(c), F(c))$

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 $V = Set$ where set is equipped with its cartesian monoidal structure.

Recall the definition of the end over

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

as an equalizer: it is the universal subobject

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

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

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

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

$\array{ F(c) &\stackrel{\eta_c}{\to} & G(c) \\ \downarrow^{F(f)} && \downarrow^{G(f)} \\ F(d) &\stackrel{\eta_d}{\to} & G(d) }$

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 $G$.

## 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)