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Idea

In the context of $V$-enriched category theory (with $V$ any suitable cosmos for enrichment) for $\mathbf{C}$, $\mathbf{D}$ a pair of $V$-enriched categories , there is, fist first of all, an ordinarycategory whose

• objects are the enriched functors $F \colon \mathbf{C} \longrightarrow \mathbf{D}$,

• morphisms $\eta \colon F \Rightarrow F'$ are the enriched natural transformations.

This notion is a generalization of the plain notion of the functor category between locally small categories, to which it reduces in the case that $V \equiv$ Set equipped with its cartesian monoidal-structure.

As such one may and does call this ordinary category the “enriched functor category” between $\mathbf{C}$ and $\mathbf{D}$, in the sense of the “category of enriched functors”.

However, this category of enriched functors canonically enhances further to a $V$-enriched category itself, hence to an “enriched-functor enriched-category”, then traditionally often denoted $[\mathbf{C}, \mathbf{D}]$ or similar.

Namely, for $F, G \,\colon\, \mathbf{C} \longrightarrow \mathbf{D}$ a pair of $V$-enriched functors between $V$-enriched categories, the collection of enriched natural transformations from $F$ to $G$ can also be given the structure of an object in $V$, in a compatible way:

Definition

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

• objects are the $V$-enriched functors $F \colon \mathbf{C} \longrightarrow \mathbf{D}$

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

  $$[\mathbf{C},\mathbf{D}](F,G) \;\coloneqq\; \textstyle{\int}_{c \in \mathbf{C}} \mathbf{D}\big(F(c),\, G(c)\big)$$

  over the enriched hom-functor

  $$\mathbf{D}\big( F(-) ,\, G(-) \big) \;\colon\; \mathbf{C}^{op} \otimes \mathbf{C} \longrightarrow V \,,$$

• the composition operation

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

(where $E_c \colon [\mathbf{C},\mathbf{D}](F,G) \to \mathbf{D}\big(F(c),G(c)\big)$ denotes the canonical morphism out of the end, i.e. the counit)

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

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

(For more see the example 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

as an equalizer: it is the universal subobject

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 $\eta \in \int_{c \in \mathbf{C}} \mathbf{D}\big(F(c), G(c)\big)$ is a collection of morphisms

such that the “left and right action” (in the sense of distributors) of $\mathbf{D}\big(F(-),G(-)\big)$ 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 $f \colon c \to d$ in $C$ the diagram

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

Pointwise order

For categories enriched in truth values, the enriched functor category is given by the pointwise order.

Properties

As internal hom in $V$-Cat

If the enriching cosmos $(V, \otimes)$ is (closed, complete and) symmetric monoidal then forming the $V$-enriched functor category out of any $V$-enriched category $\mathcal{C}$ is right adjoint to forming the enriched product category with $\mathcal{C}$, hence serves as the internal hom in $V Cat$.

[[Kelly (1982), §2.3](#Kelly82)]

Enhanced enrichment

It happens that a $V$-enriched functor category $Func(\mathbf{X},\mathbf{C})$ — which by the above discussion is a priori a $\mathbf{V}$-enriched category — carries an enhanced enrichment over the functor category $Func(\mathbf{X},\mathbf{V})$.

A common kind of example are categories of simplicial objects in an ordinary $Set$-enriched category – hence functor categories $Func(\Delta^{op}, C)$ on the opposite simplex category – which one wants to regard not just enriched in $Set$, but as enriched in $Func(\Delta^{op}, Set) \,\equiv\,$ sSet, namely in simplicial sets.

We may bootstrap the discussion of this situation by first considering the case where $\mathbf{C} = \mathbf{V}$ itself, where it means that $Func(\mathbf{X},\mathbf{V})$ is closed monoidal under the $\mathbf{X}$-objectwise tensor product taken in $\mathbf{V}$. In the base case where $\mathbf{V} =$ Set this reduces to the statement of the closed monoidal structure on presheaves.

\begin{proposition} \label{EnhancedEnichmentFormulas} Let

• $(V, \otimes)$ be a bicomplete closed monoidal category, canonically regarded as a $V$-enriched $\mathbf{V}$

  via its internal hom $[-,-]$;

• $\mathbf{C}$ a bicomplete $V$-enriched category which is

  also $V$-tensored, via $\mathbf{V} \times \mathbf{C} \overset{(\text{-})\cdot(\text{-})}{\longrightarrow} \mathbf{C}$,

  and $V$-cotensored via $\mathbf{V}^{op} \times \mathbf{C} \overset{(\text{-})^{(\text{-})}}{\longrightarrow} \mathbf{C}$;

• $\mathbf{X}$ a small $V$-enriched category.

Then the enriched functor category…

1. …$\mathbf{V}^{\mathbf{X}} \,\coloneqq\, Func(\mathbf{X},\,\mathbf{V})$ carries closed monoidal category structure

   given by the $\mathbf{X}$-objectwise tensor product

   $$\mathcal{S}_{\mathbf{X}} ,\, \mathcal{T}_{\mathbf{X}} \,\in\, \mathbf{V}^{\mathbf{X}} \;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\; \mathcal{S}_{\mathbf{X}} \otimes \mathcal{T}_{\mathbf{X}} \;\colon\; x \;\mapsto\; (\mathcal{S}_{\mathbf{X}} \otimes \mathcal{T}_{\mathbf{X}})_x \;\; \coloneqq \;\; \mathcal{S}_x \otimes \mathcal{T}_x$$

   and with corresponding internal hom given by:

   $$\mathcal{S}_{\mathbf{X}} ,\, \mathcal{T}_{\mathbf{X}} \,\in\, \mathbf{V}^{\mathbf{X}} \;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\; \big[ \mathcal{S}_{\mathbf{X}} ,\, \mathcal{T}_{\mathbf{X}} \big] \;\colon\; x \;\mapsto\; \big[ \mathcal{S}_{\mathbf{X}} ,\, \mathcal{T}_{\mathbf{X}} \big]_x \;\; \coloneqq \;\; \textstyle{\int}_{x' \in \mathbf{X}} \mathbf{V}\big( \mathbf{X}(x,x') \otimes \mathcal{S}_{x'} ,\, \mathcal{T}_{x'} \big) \,.$$

2. … $\mathbf{C}^{\mathbf{X}}$ becomes enriched, tensored and cotensored over $\mathbf{V}^{\mathbf{X}}$, via the following end-formulas, respectively:

   $$\begin{array}{rccl} \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \;\in\; \mathbf{C}^{\mathbf{X}} &\;\;\;\vdash\;\;\;& \mathbf{C}^{\mathbf{X}} \big( \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big) & \coloneqq\; \Big( x \,\mapsto\, \textstyle{\int}_{x' \in \mathbf{X}} \mathbf{C}\big( \mathbf{X}(x,x') \cdot \mathscr{V}_{x'} ,\, \mathscr{W}_{x'} \big) \Big) \;\;\; \in \; \mathbf{V}^{\mathbf{X}} \\ \mathcal{S}_{\mathbf{X}} \,\in\, \mathbf{V}^{\mathbf{X}} ;\; \mathscr{V}_{\mathbf{X}} \in\, \mathbf{C}^{\mathbf{X}} &\vdash& \mathcal{S}_{\mathbf{X}} \cdot \mathscr{V}_{\mathbf{X}} & \coloneqq\; \big( x \,\mapsto\, \mathcal{S}_{x'} \cdot \mathscr{V}_{x'} \big) \;\;\; \in \; \mathbf{C}^{\mathbf{X}} \\ \mathcal{S}_{\mathbf{X}} \,\in\, \mathbf{V}^{\mathbf{X}} ;\; \mathscr{V}_{\mathbf{X}} \in\, \mathbf{C}^{\mathbf{X}} &\vdash& \big(\mathscr{V}_{\mathbf{X}}\big)^{ \mathcal{S}_{\mathbf{X}} } & \coloneqq\; \Big( x \,\mapsto\, \textstyle{\int}_{x' \in \mathbf{X}} \big(\mathscr{W}_{x'}\big)^{ \mathbf{X}(x,x') \otimes \mathcal{S}_{x'} } \Big) \;\;\; \in \; \mathbf{C}^{\mathbf{X}} \mathrlap{\,.} \end{array}$$

\end{proposition} \begin{proof} Observe the following sequence of natural isomorphisms:

where we used, apart from the above definitions:

1. that internal hom-functors preserve limits

2. the Fubini theorem for ends

3. the co-Yoneda lemma.

This establishes that $[-,-]$ is an internal hom in $Func(\mathbf{X},\mathbf{V})$ for the objectwise tensor product, as claimed.

The natural isomorphisms needed to exhibit the (co)tensoring of $\mathbf{C}^{\mathbf{X}}$ both follow essentially the same sequence of steps, just up to the relevant substitutions. For definiteness we spell it out again, but now proceeding in the reverse order of steps:

Given

• $\mathcal{S}_{\mathbf{X}} \in Func(\mathbf{X},\mathbf{V})$,

• $\mathscr{V}_{\mathbf{X}},\, \mathscr{W}_{\mathbf{X}} \in Func(\mathbf{X},\mathbf{V})$,

• $x \in \mathbf{X}$,

consider the following sequence of $V$-enriched natural isomorphisms in these variables:

Starting with

we invoke the co-Yoneda lemma, either to introduce a fresh variable on $\mathbf{X}(x,-) \cdot \mathcal{S}_{(-)}$

or to get a fresh variable on $\mathbf{X}(x,-) \cdot \mathscr{V}_{(-)}$:

By naturality in $x \in \mathbf{X}$, these constitute the required $\mathbf{V}^{\mathbf{X}}$--enriched natural isomorphisms for exhibiting the claimed (co)tensoring:

From this, finally, follows an $\mathbf{V}^{\mathbf{X}}$-enriched composition operation

to be defined as the adjunct (via the just established adjunctions) of the consecutive evaluation maps (being themselves the respective adjunction counits):

\end{proof}

Related concepts

• enriched Reedy category

• model structure on homotopical presheaves

• model structure on sSet-presheaves

References

An original article:

• Brian Day, Max Kelly, Enriched functor categories, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106 (1969) 178-191 [[doi:10.1007/BFb0059146](https://doi.org/10.1007/BFb0059146), pdf]

Review includes

• Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf)

• Max Kelly, section 2.2 p. 29 of: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), tac:tr10, pdf]