Enriched category theory
In the context of -enriched category theory, for two -enriched functors between -enriched categories, the collection of natural transformations from to can also be given the structure of an object in , so that the functor category, denoted in the enriched context, is itself a -enriched category.
For and -enriched categories, there is a -enriched category denoted whose
objects are the -functors
hom-objects between -functors are given by the -enriched end
over the functor
Write in the following for the canonical morphism out of the end (the counit).
the composition operation
is the universal morphism into the end obtained from observing that the composites
form an extraordinary -natural family, equivalently that
equalizes the two maps appearing in the equalizer definition of the end.
Ordinary functor categories
To understand the role of the end here, it is useful to spell this out for the case where Set, where we are dealing with ordinary locally small categories.
So let 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 between the images of objects in under the functors and . So one element is a collection of morphisms
such that the “left and right action” (in the sense of distributors) of on these elements coincides. Unwrapping what this action is (see the details at end) one find that
the “right action” by a morphism is the postcomposition
the “left action” by a morphism is the precomposition .
So the invariants under the combined action are those for which for all in the diagram
commutes. Evidently, this means that the elements of the end are precisely the natural transformations between and .
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.
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)