enriched functor category



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


For CC and DD VV-enriched categories, there is a VV-enriched category denoted [C,D][C,D] whose

  • objects are the VV-functors F:CDF : C \to 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)) [C,D](F,G) := \int_{c \in C} D(F(c), G(c))

    over the functor

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

    Write in the following E c:[C,D](F,G)D(F(c),G(c))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

    K,F,G:[C,D](F,G)[C,D](K,F)[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)[C,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)) [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 VV-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)) [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), G(c))

    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. (See the examples 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 D(F(-),G(-)) : C^{op} \otimes C \to Set

as an equalizer: it is the universal subobject

cCD(F(c),G(c)) cObj(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 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 C} D(F(c), G(c)) is a collection of morphisms

(F(c)η cG(c)) cObj(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())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 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 : 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}{\to} & G(c) \\ \downarrow^{F(f)} && \downarrow^{G(f)} \\ F(d) &\stackrel{\eta_d}{\to} & G(d) }

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


An original article is

  • Brian Day Max Kelly, Enriched functor categories, Reports of the Midwest Category Seminar III (Lecture Notes in Mathematics, Volume 106, 1969), 178 - 191.

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 (pdf)

Last revised on June 8, 2018 at 02:26:11. See the history of this page for a list of all contributions to it.