This article is about ends (and coends) in category theory. For ends in topology, see at end compactification.
Enriched category theory
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
An end is a special kind of limit over a functor of the form (sometimes called a bifunctor).
If we think of such a functor in the sense of profunctors as encoding a left and right action on the object
then the end of the functor picks out the universal subobject on which the left and right action coincides. Dually, the coend of is the universal quotient of that forces the two actions of on that object to be equal.
A classical example of an end is the -object of natural transformations between -enriched functors in enriched category theory. Perhaps the most common way in which ends and coends arise is through homs and tensor products of (generalized) modules, and their close cousins, weighted limits and weighted colimits. These concepts are fundamental in enriched category theory.
In ordinary category theory
In ordinary category theory, given a functor , an end of in is an object of equipped with a universal extranatural transformation from to . This means that given any extranatural transformation from an object of to , there exists a unique map which respects the extranatural transformations.
In more detail: the end of is traditionally denoted , and the components of the universal extranatural transformation,
are called the projection maps of the end. Then, given any extranatural transformation with components
there exists a unique map such that
for every object of .
The notion of coend is dual to the notion of end. The coend of is written , and comes equipped with a universal extranatural transformation with components
We unwrap the definition of an extranatural transformation to obtain a more explicit description of an end.
Let be a functor. A wedge is an object and maps for each , such that given any morphism , the following diagram commutes:
Given a wedge and a map , we obtain a wedge by composition. Then we define the end as follows:
Let be a functor. An end of is a universal wedge, ie. a wedge such that any other wedge factors through via a map .
Dually, a cowedge is given by maps satisfying similar commutativity conditions, and a coend is a universal cowedge.
In enriched category theory
There is a definition of end in enriched category theory, as follows.
End of -valued functors
Let be a symmetric monoidal category, and let be a -enriched category. Assuming is also closed monoidal, may be considered as -enriched; in that case, suppose is a -enriched functor.
Then in particular there is a covariant action of on , with components
(where is customary notation for the hom-object of in ), and a contravariant action of on , with components
In detail, the covariant action is the adjunct of the morphism
under the Hom-adjunction
in . Similarly for the contravariant action.
A - extranatural transformation
from to consists of a family of arrows in ,
indexed over objects of , such that for every pair of objects in , the composites of (1) and (2) agree:
A -enriched end of is an object of equipped with a -extranatural transformation
such any -extranatural transformation from to is obtained by pulling back the components of along , for some unique map . That is,
for all objects of .
End of -valued functors for
If is any -enriched category and is a -enriched functor, then the end of in is, if it exists, an object of that represents the functor
That means that the end comes equipped with an -indexed family of arrows
in , such that for every object of , the family of maps
are the projection maps realizing as the corresponding end in .
End as an equalizer
Ordinary ends as equalizers
Now we motivate and define the end in enriched category theory in terms of equalizers.
Recall from the discussion at the end of limit that the limit over an (ordinary, i.e. not enriched) functor
is given by the equalizer of
If we want to generalize an expression like this to enriched category theory the explicit indexing over the set of morphisms has to be replaced by something that makes sense in an enriched category.
To that end, observe that we have a canonical isomorphism (of sets, still)
If we write for the hom-set instead
with the internal hom in Set, then the expression starts to make sense in any -enriched category.
Still equivalently but suggestively rewriting the above, we now obtain the limit over as the equalizer of
where in components
is the adjunct of
(with the last map the adjunct of ) and where
is the adjunct of
So for definiteness, the equalizer we are looking at is that of
This way of writing the limit clearly suggests that it is more natural to have and on equal footing. That leads to the following definition.
Enriched ends over -valued functors as equalizers
For a symmetric monoidal category, a -enriched category and a -enriched functor, the end of is the equalizer
with in components given by
being the adjunct of
being the adjunct of
This definition manifestly exhibits the end as the equalizer of the left and right action encoded by the distributor .
Dually, the coend of is the coequalizer
with the parallel morphisms again induced by the two actions of .
End as a weighted limit
The end for -functors with values in serves, among other things, to define weighted limits, and weighted limits in turn define ends of bifunctors with values in more general -categories.
For and both -categories and an -enriched functor, the end of is the weighted limit of
with weight . The coend of is the colimit
of weighted by the hom functor of .
Connecting the two definitions
If is a -category, then the hom functor is the coequalizer in
It is also a general fact (see e.g. Kelly, ch. 3) that weighted (co)limits are cocontinuous in their weight: that is,
This implies that takes the coequalizer above to an equalizer, which, after some fiddling with the Yoneda lemma, turns out to be isomorphic to (1). Similarly, takes the analogous coequalizer presentation of to (2).
-enriched coends as ordinary colimits
Let the enriching category be Set. We describe a special way in this case to express ends/coends that give weighted limits/colimits in terms of ordinary (co)limits over categories of elements.
We have a natural isomorphism in
between the coend as indicated and the colimit over the opposite of the category of elements of .
This is equation (3.34) in (Kelly) in view of (3.70).
Any continuous functor preserves ends, and any cocontinuous functor preserves coends. In particular, for functors and , we have the isomorphisms
If is a representable functor, then
is the over category over the representing object . This has a terminal object, namely ). Therefore
Since this is natural in , the above proposition asserts a natural isomorphism
This statement is sometimes called the co-Yoneda lemma.
Commutativity of ends and coends
Ordinary limits commute with each other, if both limits exist separately. The analogous statement does hold for ends and coends. Since there it looks like the commutativity of two integrals, it is called the Fubini theorem for ends (for instance Kelly, p. 29).
(Fubini theorem for ends)
Let be a symmetric monoidal category. Let and be small -enriched categories.
be a -enriched functor. Then:
If for all object the end exists, then
if either side exists. In particular, since this implies that
if either side exists.
Let be functors between two categories, and let be the set of natural transformations between them. Then we have
An element of is by definition a collection of morphisms in such that for any morphism in , the following square commutes:
which is by definition a natural transformation .
Enriched functor categories
In light of Proposition 3, we can define the natural transformations object for enriched functors as an end:
For and both -enriched categories, the -enriched functor category is the -enriched category whose
objects are -enriched functors ;
hom-objects in are given by the end-formula .
For this reproduces of course the ordinary functor category.
If the -enriched category is tensored over , then the (left) Kan extension of a functor along a functor is given by the coend
See Kan extension for more details.
A special case of the example of Kan extension is that of geometric realization.
Very generally, geometric realization is the left Kan extension of a functor along the Yoneda embedding .
The “geometric realization” of an object with respect to is then the coend
where the last step on the right uses the Yoneda lemma.
More specifically, traditionally this is thought of as applying to the case where is the simplex category and where regards the abstract -simplex as the standard simplex as a topological space.
Tensor product of functors
If and are functors, their tensor product is the coend
where the tensor product on the right hand side refers to some monoidal structure on .
The formal properties of (co)ends in Propositions 1, 2 and 3 allow us to prove certain results by abstract nonsense.
Let be a functor. We prove the co-Yoneda lemma, that
We perform the following manipluations, where each isomorphism is natural:
So by the Yoneda lemma, we have
More examples can be found in Fosco.
The standard reference is
Max Kelly, Basic concepts in enriched category theory (pdf)
ends of -valued bifunctors are discussed in section 2.1
the enriched functor category that they give rise to is discussed in section 2.2;
enriched weighted limits in terms of enriched functor categories are in section 3.1
the end of general -enriched functors in terms of weighted limits is in section 3.10 .
Ends, -Category Café discussion.
Fosco Loregian, This is the (co)end, my only (co)friend (arXiv).