Generalized Yoneda Lemma (from Pat Donaly)

Generalized Yoneda Lemma (from Pat Donaly)

[Jpdonaly]

To all category theorists:

Some time ago I noticed a generalization of the Yoneda Lemma which runs as
follows: As in the classical version, one begins with a nonvoid category A and a
pair of function-valued bifunctors on the product of the following
categories: The first is the category of function-valued natural transformations on A,
and the second is A itself.  One of the bifunctors is the evaluation functor
(t,a)-->t(a), where t is a function-valued natural transformation on A, and a is
an A-morphism. Call this bifunctor E. The other bifunctor Y is described as
an iterated Yoneda embedding which has been unpartialled, and the classical
Yoneda Lemma states that Y is naturally isomorphic to E.  Also, a particular
isomorphism is specified, at least on objects.  But the situation is clarified by
observing that there is a bijective parameterization of the homset H of
natural transformations which entwine Y with E by what has to be called the center
of A, that is, by the commutative monoid C of natural transformations (under
(indifferently) pointwise or function composition) which entwine the identity
functor on A with itself.  This parameterization sends natural isomorphisms to
natural isomorphisms, and, applying it to the identity functor on A gives the
classical Yoneda Lemma, but, of course, there might be other isomorphisms in C,
hence other isomorphisms of Y with E.

The parameterization formula from the center C into the homset H is not
particularly enlightening, but its inverse is literally the formula which expresses
an adjunctional unit in terms of its adjunction (regarded as a
function-valued natural bitransformation). In fact, this development all comes out of the
observation that the adjunctional unit formula extends to a concept which is
well beyond the idea of an adjunction, and this generalized unit concept even
includes naturality in a certain sense. Thus one nearly has naturality, the
Yoneda Lemma, the adjunctional unit concept and the idea of the center of a
category all together in one basket.

It is my personal opinion that these facts are most easily seen by working
with natural transformations in terms of their fully extended entwining
functions (so that "horizontal" composition is function composition and "vertical"
composition is pointwise composition), but ruthless experience tells me that
someone who is not familiar with the journal literature should take care not to
underestimate a priori the mysterious capabilities of diagrammers, who, after
all, created category theory in the first place.  Moreover, I have no way to
assess the value of this generalization in terms of the classical applications of
the Yoneda Lemma, as in, say, the calculation of characteristic classes. Thus
I am interested in getting directions to relevant portions of the literature,
and would someone point out any applications of this generalized Yoneda
Lemma?

Pat Donaly