The term Yoneda reduction is was coined by Todd Trimble in his (unpublished) thesis. It refers to a technique based on the Yoneda lemma for performing a number of end and coend calculations which arise in coherence theory and enriched category theory.
There are various formulations of the Yoneda lemma. One says that given a presheaf , there is a canonical isomorphism
where “Nat” refers to the set of natural transformations between presheaves ; in other words, the hom
appropriate to the presheaf category.
There is an -enriched category version, whenever is a category enriched in a complete, cocomplete, symmetric monoidal closed category . Here “Nat” is constructed as an enriched end (an example of a weighted limit):
and therefore the enriched Yoneda lemma gives an isomorphism
which is (-)natural in ; we may therefore write
and this isomorphism is -natural in .
We pause to give an instance of the Yoneda lemma which is both familiar and which serves to inform much of the module-theoretic terminology in the discussion below. Let ; let be a ring (conceived as an -enriched category with exactly one object ). Then is the (-enriched) category of right -modules, or equivalently, left -modules). The presheaf is just the underlying abelian group of seen as a right module over the ring , also known as the regular representation.
The first formulation (1) of the Yoneda lemma would simply say that at the level of abelian groups, we have for any right -module
Further taking into account the “naturality” in the argument bullet, the formulation (2) says that actually we have an isomorphism at the level of right -modules
where the module structure on the right side arises by considering the argument now as a bimodule over the (ring) .
The (enriched) Yoneda lemma is nothing but a far-reaching extrapolation of this basic isomorphism: it says
where the -presheaf or right -module hom on the right is appropriately constructed as an enriched end, and is a treated as a -enriched “bimodule” over , and plays the role of the “regular representation” of .
In the first place, given a ring , there is a familiar monoidal category of -bimodules (and bimodule morphisms). If are bimodules over , with left -actions denoted by ’s and the right actions by ’s, their tensor product , defined by the coequalizer
(where the two parallel arrows are , ) carries an evident -bimodule structure. Each of the functors and admits a right adjoint expressed by natural isomorphisms of abelian groups
where denotes the abelian group of left -module maps , equipped with its natural -bimodule structure; is similar. Thus the monoidal category of -bimodules is biclosed.
More generally, there is a bicategory whose objects or 0-cells are rings , and whose morphisms or 1-cells are left -, right -bimodules. 2-cells are homomorphisms of bimodules. If and are bimodules, then their bimodule composite is . This too is a biclosed bicategory, meaning that
This generalized module theory can be pursued much further.
(Lost a bunch of work, due to vagaries of computers. Sigh. Will return later.)
If is symmetric monoidal, then the monoid of endomorphisms on the -fold tensor functor
is in bijection with the monoid of endomorphisms on the unit object .
By fully and faithfully embedding (as a symmetric monoidal category) into , we may without loss of generality suppose is complete, cocomplete, symmetric monoidal closed.
The result is by induction on : observe that a map
natural in all the arguments , in particular in , corresponds to a map dinatural in :
and hence to a map to the end
where the end exists and is isomorphic to
by Yoneda reduction. This completes the induction.
(It’s been ages since I’ve thought about this. I need to think through the argument carefully again.)