Note: for a category , we write
for the hom-class of maps from
to
.
An adjunction is a pair of functors and
with a certain relationship. There are many texts expositing adjunctions, but I have not seen one covering what I think of as their fundamental theorem. This is regrettable, since I think many people would appreciate the simplifying effect this result has on the theory of adjunctions in
. My goal today is to cover this theorem using coend calculus.
Adjunctions in a nutshell. First, an example. Let be the category of sets and let
be the category of
-modules for a ring
. There are functors
sending a set
to
and a functor
sending an
-module
to its underlying set. There is a natural way of going back and forth between maps of
-modules
and maps of sets
. For a map of
-modules,
, we get a map of sets by first applying
(this really does nothing) and then precomposing with a map
of sets
sending
to the element
in the underlying set of
where
and
for
. The resulting map is
. For a map of sets
, we get a map
where
sends
to
, where the sum is taken in
. What we have here is a natural correspondence between hom sets
and
. To go one way we apply
and then precompose with
(kind of like correcting for an error, since
and
are not inverses), and to go the other way we apply
and then postcompose with
. This hints at the definition of an adjoint:
Definition: Let and
be functors. We say that
is adjoint to
, or that
and
are adjoint, if there is a natural isomorphism
between the functors
.
The example also hints at what I call the fundamental theorem of adjunctions in the -category
:
Theorem: Let and
be categories and
and
be functors. Then
(i) and
are naturally isomorphic.
(ii) and
are naturally isomorphic.
Proof: These are dual; we prove the first. For the isomorphism itself, we have a series of natural isomorphisms:
This finishes the proof.
Using coend calculus like that is nice, but we are also interested in an explicit description of the isomorphisms and
! To achieve this description, notice that the following diagram commutes for each
and each
:
Take a natural transformation . For each
, define a natural transformation
such that
sends
to
, to
. Define a natural transformation
where
. The composition
gives the following diagram of mappings:
As this holds for each and each
, we have
.
Next take a natural transformation . For each
, define a natural transformation
such that
. Define a natural transformation
such that
. The composition
gives the following diagram of mappings:
As this holds for each and each
,
.
Hence sends a natural transformation
to the natural transformation
where
for
,
, and
, and
sends a natural transformation
to the natural transformation
such that
.
This theorem puts us in a place where we can prove other results about adjoints much more easily:
Theorem: The following are equivalent:
(i) The functors and
are naturally isomorphic.
(ii)(a) (Universal Morphisms Definition) There is a natural transformation such that, for each
, each
, and each morphism
in
, there is a unique morphism
in
such that
.
(ii)(b) (Universal Morphisms Definition) There is a natural transformation such that, for each
, each
, and each
in
, there is a unique morphism
in
such that
.
(iii) (The Unit Counit Definition) There are natural transformations and
such that the following compositions are the identity morphism, for each
and
.
Proof: . Suppose that the functors
and
are naturally isomorphic, and take a natural isomorphism
. Let
be the corresponding natural transformation. Take
,
, and
. There is a unique
such that
, since
is a natural isomorphism. But
by the fundamental theorem. This shows (ii)(a).
. Suppose that there is a natural transformation
such that, for each
, each
, and each morphism
in
, there is a unique morphism
in
such that
. Let
be the natural transformation corresponding to
. Take
and
.
for each
, so that, for each
, there is a unique
such that
. This shows (i).
. Suppose that the functors
and
are naturally isomorphic, and take a natural isomorphism
. Let
be the corresponding natural transformation. Take
,
, and
. There is a unique
such that
, since
is a natural isomorphism. But
by the fundamental theorem. This shows (ii)(b).
. Suppose that there is a natural transformation
such that, for each
, each
, and each morphism
in
, there is a unique morphism
in
such that
. Let
be the natural transformation corresponding to
. Take
and
.
for each
, so that, for each
, there is a unique
such that
. This shows (i).
. Suppose that the functors
and
are naturally isomorphic, and take a natural isomorphism
. Let
be the natural transformation corresponding to
and let
be the natural transformation corresponding to
. For
, we have
and
. Suppose
holds. Let
be the natural transformation induced by
, and let
be the natural transformation induced by
.
Take . Write
for the induced natural transformation. The following diagram commutes by the Yoneda lemma:
So . So
is the identity natural transformation. Hence $EH$ is the identity natural transformation.
Take . Write
for the induced natural transformation. The following diagram commutes by the Yoneda lemma:
So . So
is the identity natural tarnsformation. Hence
is the identity natural transformation.
It follows that and
are inverse natural transformations, and therefore natural isomorphisms. This shows (i).