nLab enriched functor category




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

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 VV \equiv Set equipped with its cartesian monoidal-structure.

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

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

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


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

  • objects are the VV-enriched functors F:CDF \colon \mathbf{C} \longrightarrow \mathbf{D}

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

    [C,D](F,G) cCD(F(c),G(c)) [\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

    D(F(),G()):C opCV, \mathbf{D}\big( F(-) ,\, G(-) \big) \;\colon\; \mathbf{C}^{op} \otimes \mathbf{C} \longrightarrow V \,,
  • the composition operation

K,F,G:[C,D](F,G)[C,D](K,F)[C,D](K,G) \circ_{K,F,G} \,\colon\, [\mathbf{C},\mathbf{D}](F,G) \otimes [\mathbf{C},\mathbf{D}](K,F) \longrightarrow [\mathbf{C},\mathbf{D}](K,G)

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

[C,D](F,G)[C,D](K,F)E cE dD(F(c),G(c))D(K(c),F(c)) K(c),F(c),G(c)D(K(c),F(c)) [\mathbf{C},\mathbf{D}](F,G) \otimes [\mathbf{C},\mathbf{D}](K,F) \stackrel{E_c\otimes E_d}{\longrightarrow} \mathbf{D}\big(F(c),G(c)\big) \otimes \mathbf{D}\big(K(c),F(c)\big) \stackrel{\circ_{K(c), F(c), G(c)}}{\longrightarrow} \mathbf{D}\big(K(c), F(c)\big)

(where E c:[C,D](F,G)D(F(c),G(c))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 V -natural family, equivalently that

[C,D](F,G)[C,D](K,F) cObj(c)E cE c cObj(c)D(F(c),G(c))D(K(c),F(c)) cObj(c) K(c),F(c),G(c) cObj(c)D(K(c),G(c)) [\mathbf{C},\mathbf{D}](F,G) \otimes [\mathbf{C},\mathbf{D}](K,F) \stackrel {\prod_{c \in Obj(c)}E_c\otimes E_c} {\longrightarrow} \prod_{c \in Obj(c)} \mathbf{D}\big(F(c),G(c)\big) \otimes \mathbf{D}\big(K(c),\,F(c)\big) \stackrel{\prod_{c \in Obj(c)} \circ_{K(c), F(c), G(c)}}{\longrightarrow} \prod_{c \in Obj(c)} \mathbf{D}\big(K(c),\,G(c)\big)

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


For V=V = Set, so that VV-enriched categories are just ordinary locally small categories, the VV-enriched functor category coincides with the ordinary functor category.

(For more see the example below.)


Ordinary functor categories

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

So let V=SetV = Set where set is equipped with its cartesian monoidal structure.

Recall the definition of the end over

D(F(),G()):C opCSet \mathbf{D}\big( F(-),G(-) \big) \,\colon\, \mathbf{C}^{op} \otimes \mathbf{C} \longrightarrow Set

as an equalizer: it is the universal subobject

cCD(F(c),G(c)) cObj(C)D(F(c),G(c)) \textstyle{\int}_{c \in C} \mathbf{D}\big(F(c),\, G(c)\big) \hookrightarrow \prod_{c \in Obj(C)} \mathbf{D}\big(F(c),\, G(c)\big)

of the product of all hom-sets in DD between the images of objects in CC under the functors FF and GG. So one element η cCD(F(c),G(c)) \eta \in \int_{c \in \mathbf{C}} \mathbf{D}\big(F(c), G(c)\big) is a collection of morphisms

(F(c)η cG(c)) cObj(c) \big( F(c) \stackrel{\eta_c}{\longrightarrow} G(c) \big)_{c \in Obj(c)}

such that the “left and right action” (in the sense of distributors) of D(F(),G())\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 cfdc \stackrel{f}{\to} d is the postcomposition (F(c)η cG(c))(F(c)η cG(c)G(f)G(d)) (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 cfdc \stackrel{f}{\to} d is the precomposition (F(d)η dG(d))(F(c)F(f)F(d)η dG(d)) (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 η\eta for which for all f:cdf \colon c \to d in CC the diagram

F(c) η c G(c) F(f) G(f) F(d) η d G(d) \array{ F(c) &\stackrel{\eta_c}{\longrightarrow} & G(c) \\ \Big\downarrow\mathrlap{{}^{F(f)}} && \Big\downarrow\mathrlap{{}^{G(f)}} \\ F(d) & \stackrel{\eta_d}{\longrightarrow} & G(d) }

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

Pointwise order

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


As internal hom in VV-Cat

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

[Kelly (1982), §2.3]

Enhanced enrichment

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

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

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



Then the enriched functor category…

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

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

    𝒮 X,𝒯 XV X𝒮 X𝒯 X:x(𝒮 X𝒯 X) x𝒮 x𝒯 x \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:

    𝒮 X,𝒯 XV X[𝒮 X,𝒯 X]:x[𝒮 X,𝒯 X] x xXV(X(x,x)𝒮 x,𝒯 x). \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. C X\mathbf{C}^{\mathbf{X}} becomes enriched, tensored and cotensored over V X\mathbf{V}^{\mathbf{X}}, via the following end-formulas, respectively:

    𝒱 X,𝒲 XC X C X(𝒱 X,𝒲 X) (x xXC(X(x,x)𝒱 x,𝒲 x))V X 𝒮 XV X;𝒱 XC X 𝒮 X𝒱 X (x𝒮 x𝒱 x)C X 𝒮 XV X;𝒱 XC X (𝒱 X) 𝒮 X (x xX(𝒲 x) X(x,x)𝒮 x)C X. \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}


Observe the following sequence of natural isomorphisms:

Func(X,V)( X,[𝒮 X,𝒯 X]) xXV( x, yXV(X(x,y)𝒮 y,𝒯 y)) by the definitions xX yXV( x,V(X(x,y)𝒮 y,𝒯 y)) since internal homs preserve limits xX yXV(X(x,y) x𝒮 y,𝒯 y) by closed monoidal structure ofV yX xXV(X(x,y) x𝒮 y,𝒯 y) by Fubini theorem for ends yXV(( xXX(x,y) x)𝒮 y,𝒯 y) since internal homs preserve limits yXV( y𝒮 y,𝒯 y) by co-Yoneda lemma yXV(( X𝒮 X) y,𝒯 y) by definition Func(X,V)( X𝒮 X,𝒯 X) by definition, \begin{array}{ll} Func(\mathbf{X},\,\mathbf{V}) \big( \mathcal{R}_{\mathbf{X}} \;,\; [\mathcal{S}_{\mathbf{X}},\, \mathcal{T}_{\mathbf{X}}] \big) \\ \;\simeq\; \textstyle{\int}_{x \in \mathbf{X}} \mathbf{V} \Big( \mathcal{R}_{x} \;,\; \int_{y \in \mathbf{X}} \mathbf{V} \big( \mathbf{X}(x,y) \otimes \mathcal{S}_{y} \;,\; \mathcal{T}_{y} \big) \Big) & \text{by the definitions} \\ \;\simeq\; \int_{x \in \mathbf{X}} \int_{y \in \mathbf{X}} \mathbf{V} \Big( \mathcal{R}_{x} \;,\; \mathbf{V} \big( \mathbf{X}(x,y) \otimes \mathcal{S}_{y} \;,\; \mathcal{T}_{y} \big) \Big) & \text{ since internal homs preserve limits } \\ \;\simeq\; \int_{x \in \mathbf{X}} \int_{y \in \mathbf{X}} \mathbf{V} \Big( \mathbf{X}(x,y) \otimes \mathcal{R}_{x} \otimes \mathcal{S}_{y} \;,\; \mathcal{T}_{y} \Big) & \text{by closed monoidal structure of}\;\mathbf{V} \\ \;\simeq\; \int_{y \in \mathbf{X}} \int_{x \in \mathbf{X}} \mathbf{V} \Big( \mathbf{X}(x,y) \otimes \mathcal{R}_{x} \otimes \mathcal{S}_{y} \;,\; \mathcal{T}_{y} \Big) & \text{ by Fubini theorem for ends } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{V} \Big( \big( \int^{x \in \mathbf{X}} \mathbf{X}(x,y) \otimes \mathcal{R}_{x} \big) \otimes \mathcal{S}_{y} \;,\; \mathcal{T}_{y} \Big) & \text{ since internal homs preserve limits } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{V} \Big( \mathcal{R}_{y} \otimes \mathcal{S}_{y} \;,\; \mathcal{T}_{y} \Big) & \text{ by co-Yoneda lemma } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{V} \Big( \big( \mathcal{R}_{\mathbf{X}} \otimes \mathcal{S}_{\mathbf{X}} \big)_{y} \;,\; \mathcal{T}_{y} \Big) & \text{ by definition } \\ \;\simeq\; Func(\mathbf{X},\mathbf{V}) \big( \mathcal{R}_{\mathbf{X}} \otimes \mathcal{S}_{\mathbf{X}} \;,\; \mathcal{T}_{\mathbf{X}} \big) & \text{ by definition, } \end{array}

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(X,V)Func(\mathbf{X},\mathbf{V}) for the objectwise tensor product, as claimed.

The natural isomorphisms needed to exhibit the (co)tensoring of C X\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:


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

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

  • xXx \in \mathbf{X},

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

Starting with

C X(𝒮 X𝒱 X,𝒲 X) x zXC(X(x,z)(𝒮 X𝒱 X) z,𝒲 z) by definition zXC(X(x,z)𝒮 z𝒱 z,𝒲 z) by definition \begin{array}{ll} \mathbf{C}^{\mathbf{X}} \big( \mathcal{S}_{\mathbf{X}} \cdot \mathscr{V}_{\mathbf{X}} \;,\; \mathscr{W}_{\mathbf{X}} \big)_x \\ \;\simeq\; \int_{z \in \mathbf{X}} \mathbf{C}\Big( \mathbf{X}(x,z) \cdot \big( \mathcal{S}_{\mathbf{X}} \cdot \mathscr{V}_{\mathbf{X}} \big)_{z} ,\, \mathscr{W}_{z} \Big) & \text{ by definition } \\ \;\simeq\; \int_{z \in \mathbf{X}} \mathbf{C}\big( \mathbf{X}(x,z) \cdot \mathcal{S}_{z} \cdot \mathscr{V}_{z} \;,\; \mathscr{W}_{z} \big) & \text{ by definition } \\ \;\simeq\; \cdots \end{array}

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

zXC( yXX(y,z)(X(x,y)𝒮 y)𝒱 z,𝒲 z) co-Yoneda lemma zX yXC(X(y,z)X(x,y)𝒮 y𝒱 z,𝒲 z) enriched homs preserve enriched limits zX yXV(X(x,y)𝒮 y,C(X(y,z)𝒱 z,𝒲 z)) tensoring iso of C yX zXV(X(x,y)𝒮 y,C(X(y,z)𝒱 z,𝒲 z)) Fubini theorem for ends yXV(X(x,y)𝒮 y, zXC(X(y,z)𝒱 z,𝒲 z)) enriched homs preserve enriched limits yXV(X(x,y)𝒮 y,C X(𝒱 X,𝒲 X) y) definition V X(𝒮 X,C X(𝒱 X,𝒲 X)) x definition \begin{array}{ll} \cdots \\ \;\simeq\; \int_{z \in \mathbf{X}} \mathbf{C}\Big( \int^{y \in \mathbf{X}} \mathbf{X}(y,z) \cdot \big( \mathbf{X}(x,y) \cdot \mathcal{S}_{y} \big) \cdot \mathscr{V}_{z} \;,\; \mathscr{W}_{z} \Big) & \text{ co-Yoneda lemma } \\ \;\simeq\; \int_{z \in \mathbf{X}} \int_{y \in \mathbf{X}} \mathbf{C}\Big( \mathbf{X}(y,z) \cdot \mathbf{X}(x,y) \cdot \mathcal{S}_{y} \cdot \mathscr{V}_{z} \;,\; \mathscr{W}_{z} \Big) & \text{ enriched homs preserve enriched limits } \\ \;\simeq\; \int_{z \in \mathbf{X}} \int_{y \in \mathbf{X}} \mathbf{V} \Big( \mathbf{X}(x,y) \cdot \mathcal{S}_{y} \;,\; \mathbf{C}\big( \mathbf{X}(y,z) \cdot \mathscr{V}_{z} \;,\; \mathscr{W}_{z} \big) \Big) & \text{ tensoring iso of }\; \mathbf{C} \\ \;\simeq\; \int_{y \in \mathbf{X}} \int_{z \in \mathbf{X}} \mathbf{V} \Big( \mathbf{X}(x,y) \cdot \mathcal{S}_{y} \;,\; \mathbf{C}\big( \mathbf{X}(y,z) \cdot \mathscr{V}_{z} \;,\; \mathscr{W}_{z} \big) \Big) & \text{ Fubini theorem for ends } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{V} \Big( \mathbf{X}(x,y) \cdot \mathcal{S}_{y} \;,\; \int_{z \in \mathbf{X}} \mathbf{C}\big( \mathbf{X}(y,z) \cdot \mathscr{V}_{z} \;,\; \mathscr{W}_{z} \big) \Big) & \text{ enriched homs preserve enriched limits } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{V}\Big( \mathbf{X}(x,y) \cdot \mathcal{S}_{y} \;,\; \mathbf{C}^{\mathbf{X}} \big( \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big)_y \Big) & \text{ definition } \\ \;\simeq\; \mathbf{V}^{\mathbf{X}}\Big( \mathcal{S}_{\mathbf{X}} \;,\; \mathbf{C}^{\mathbf{X}} \big( \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big) \Big)_x & \text{ definition } \end{array}

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

zXC(𝒮 z yXX(y,z)(X(x,y)𝒱 y),𝒲 z) co-Yoneda lemma zX yXC(X(x,y)X(y,z)𝒮 z𝒱 y,𝒲 z) enriched homs preserve enriched limits zX yXC(X(x,y)𝒱 y,(𝒲 z) X(y,z)𝒮 z) cotensoring iso of C yX zXC(X(x,y)𝒱 y,(𝒲 z) X(y,z)𝒮 z) Fubini theorem for ends yXC(X(x,y)𝒱 y, zX(𝒲 z) X(y,z)𝒮 z) enriched homs preserve enriched limits yXC(X(x,y)𝒱 y,((𝒲 X) 𝒮 X) y) definition C X(𝒱 X,(𝒲 X) 𝒮 X) x definition \begin{array}{ll} \cdots \\ \;\simeq\; \int_{z \in \mathbf{X}} \mathbf{C} \Big( \mathcal{S}_z \cdot \int^{y \in \mathbf{X}} \mathbf{X}(y,z) \cdot \big( \mathbf{X}(x,y) \cdot \mathcal{V}_{y} \big) \;,\; \mathscr{W}_{z} \Big) & \text{ co-Yoneda lemma } \\ \;\simeq\; \int_{z \in \mathbf{X}} \int_{y \in \mathbf{X}} \mathbf{C} \Big( \mathbf{X}(x,y) \cdot \mathbf{X}(y,z) \cdot \mathcal{S}_z \cdot \mathcal{V}_{y} \;,\; \mathscr{W}_{z} \Big) & \text{ enriched homs preserve enriched limits } \\ \;\simeq\; \int_{z \in \mathbf{X}} \int_{y \in \mathbf{X}} \mathbf{C} \Big( \mathbf{X}(x,y) \cdot \mathcal{V}_{y} \;,\; \big(\mathscr{W}_{z}\big)^{ \mathbf{X}(y,z) \cdot \mathcal{S}_z } \Big) & \text{ cotensoring iso of }\; \mathbf{C} \\ \;\simeq\; \int_{y \in \mathbf{X}} \int_{z \in \mathbf{X}} \mathbf{C} \Big( \mathbf{X}(x,y) \cdot \mathcal{V}_{y} \;,\; \big(\mathscr{W}_{z}\big)^{ \mathbf{X}(y,z) \cdot \mathcal{S}_z } \Big) & \text{ Fubini theorem for ends } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{C} \Big( \mathbf{X}(x,y) \cdot \mathcal{V}_{y} \;,\; \int_{z \in \mathbf{X}} \big(\mathscr{W}_{z}\big)^{ \mathbf{X}(y,z) \cdot \mathcal{S}_z } \Big) & \text{ enriched homs preserve enriched limits } \\ \;\simeq\; \int_{y \in \mathbf{X}} \mathbf{C} \bigg( \mathbf{X}(x,y) \cdot \mathcal{V}_{y} \;,\; \Big( \big( \mathscr{W}_{\mathbf{X}} \big)^{ \mathcal{S}_{\mathbf{X}} } \Big)_y \bigg) & \text{ definition } \\ \;\simeq\; \mathbf{C}^{\mathbf{X}} \Big( \mathcal{V}_{\mathbf{X}} \;,\; \big(\mathscr{W}_{\mathbf{X}}\big)^{ \mathcal{S}_{\mathbf{X}} } \Big)_x & \text{ definition } \end{array}

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

C X(𝒮 X𝒱 X,𝒲 X)V X(𝒮 X,C X(𝒱 X,𝒲 X))C X(𝒱 X,(𝒲 X) 𝒮 X). \mathbf{C}^{\mathbf{X}} \big( \mathcal{S}_{\mathbf{X}} \cdot \mathscr{V}_{\mathbf{X}} \;,\; \mathscr{W}_{\mathbf{X}} \big) \;\simeq\; \mathbf{V}^{\mathbf{X}} \Big( \mathcal{S}_{\mathbf{X}} \;,\; \mathbf{C}^{\mathbf{X}}\big( \mathscr{V}_{\mathbf{X}} \;,\; \mathscr{W}_{\mathbf{X}} \big) \Big) \;\simeq\; \mathbf{C}^{\mathbf{X}} \Big( \mathscr{V}_{\mathbf{X}} \;,\; \big(\mathscr{W}_{\mathbf{X}}\big)^{ \mathcal{S}_{\mathbf{X}} } \Big) \,.

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

C X( X,𝒱 X)C X(𝒱 X,𝒲 X)C X( X,𝒲 X) \mathbf{C}^{\mathbf{X}}\big( \mathscr{R}_{\mathbf{X}} ,\, \mathscr{V}_{\mathbf{X}} \big) \otimes \mathbf{C}^{\mathbf{X}}\big( \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big) \longrightarrow \mathbf{C}^{\mathbf{X}}\big( \mathscr{R}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big)

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

XC X( X,𝒱 X)C X(𝒱 X,𝒲 X)ev𝒱 XC X(𝒱 X,𝒲 X)ev𝒲 X. \mathscr{R}_{\mathbf{X}} \cdot \mathbf{C}^{\mathbf{X}}\big( \mathscr{R}_{\mathbf{X}} ,\, \mathscr{V}_{\mathbf{X}} \big) \cdot \mathbf{C}^{\mathbf{X}}\big( \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big) \overset{ev}{\longrightarrow} \mathscr{V}_{\mathbf{X}} \cdot \mathbf{C}^{\mathbf{X}}\big( \mathscr{V}_{\mathbf{X}} ,\, \mathscr{W}_{\mathbf{X}} \big) \overset{ev}{\longrightarrow} \mathscr{W}_{\mathbf{X}} \,.


An original article:

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, tac:tr10, pdf]

Last revised on April 2, 2024 at 19:09:16. See the history of this page for a list of all contributions to it.