This post is part of a series on category theory, see Overview of Blog Posts for all posts. All categories are assumed to be locally small.
Introduction
In this post, we shall establish different perspectives on the previously established concept of adjoint functors, along with more examples and properties. We shall use the same idea that underlies the Yoneda lemma to characterize an adjunction using special natural transformations called unit and counit.
To get a class of examples for adjunction, we shall study free objects in concrete categories. (Where “concrete” is a technical term. But don’t worry, the concept of concrete categories will be made concrete in examples of concrete concrete categories.).
Through them, we shall see how a natural generalization of the concept of a basis from linear algebra leads to adjunctions.
By following this approach, we will also see that there is a close connection between adjoint functors and representability.
Unit and Counit
An adjunction between two functors is given by a natural isomorphism between two functors involving Hom-functors. We know from the Yoneda lemma and in particular from its proof that a natural transformation from a Hom functor is determined by its action on identities. In the same spirit, all the information of an adjunction is contained in the action on certain identity morphisms.
Let and
be categories and let
be functors. Suppose we are given an adjunction, i.e. a natural isomorphism
. Then for each
and each
, we have a bijection
. If we fix
and set
, then we obtain an element
. Similarly, we can fix
and set
and obtain an element
.
Definition C9.1 In the above discussion, is called the unit of the adjunction (at
) and
is called the counit of the adjunction.
Lemma C9.2 The collection of morphisms is natural in
.
Proof Let be a morphism in
, then we must show that the following diagram commutes:
i.e. .
Via naturality of in the second argument, we obtain a commutative diagram:
Applying this to the identity yields the equation
, i.e.
.
Naturality of in the first argument yields another commutative diagram:
Applying this to the identity leads to the equation
, i.e.
.
We have thus shown that , as both sides equal
.
This lemma shows that the unit of an adjunction defines a natural transformation, which we denote by .
A very similar proof leads to the following lemma:
Lemma 9.3 Given the established context, the collection of morphisms is natural in
.
So far, we have established how an adjunction gives rise to two natural transformations
and
. Following the Yoneda philosophy as described in the beginning of this section, our goal is to show that this pair of natural transformations already encapsulates the whole adjunction. To this end, we will prove properties which will turn out to give a necessary and sufficient condition for a pair of such natural transformations to come from an adjunction.
Lemma C9.4 We have for every ,
Proof Naturality of in the first argument respect to
yields the commutative diagram:
Applying this to , we get
, so that we obtain by the injectivity of
the desired equality
.
Lemma C9.5 We have for every ,
.
Proof Applying naturality of in the second argument with respect to
yields the following commutative diagram:
Applying this to , we obtain
, i.e.
.
Lemma C9.6 Given a pair of functors and a pair of natural transformations
,
such that for all
and
, we have
and
, the map
is a natural isomorphism with inverse
.
Proof Naturality of follows from the naturality of
. For any
and
, we obtain
. By naturality of
with respect to
, we have
. By assumption
. Combining these, we see that
. For
, we have
, which can be shown by analogous reasoning from the naturality of
with respect to
and the condition
.
Putting everything together, we have shown:
Theorem C9.7 Given two categories and
and functors
, the datum of an adjunction between
and
is equivalent to giving two natural transformations
,
satisfying the so-called unit-counit equations: for all
and
, we have
and
.
Free Objects
We shall now return from the general to the concrete (quite literally):
Definition C9.8 A concrete category is a pair where
is a category and
is a faithful functor.
Using the faithful functor , which we think of as a “forgetful” functor, we can think of objects in the category
as sets and of morphisms as maps. A lot of common examples of categories have a canonical choice for a faithful functor
.
Example C9.9 The category of sets is concrete with
the identity functor.
Examples C9.10 The categories of semigroups, monoids, groups, abelian groups, rings, commutative rings, fields, modules over a fixed base ring etc. with the respective homomorphisms as morphisms are all concrete categories where is the forgetful functor to
Examples C9.11 The category of topological spaces with continuous maps is a concrete category, so is the category of smooh manifolds with smooth maps, and so is the category of complex-analytic manifolds with holomorphic maps.
Example C9.12 Let be a monoid, considered as a one-object category. Then the idea from the proof of Cayley’s theorem from group theory allows us to realize
as a concrete category: to define the functor
, send the unique object of
to the set underlying set of
(i.e. the set of all morphisms in
) and define
to be left multiplication with
.
Example C9.13 Generalizing the last example, every small category admits the structure of a concrete category in a canonical way: consider the functor
that sends an object
to the set of all morphisms to
and a morphism
to the map
given by post-composing with
. Then
is a faithful functor, which makes
a concrete category.
Example C9.14 The category of all small categories is a concrete category. We can define a functor
, by sending a small category
to the set of all morphisms in
and sending a functor between small categories to the induced map on the sets of morphisms. (Note that the action on the objects can be recovered from the action on morphisms by looking at the images of the identities). On the other hand,the functor
that sends a small category to the set of its objects is not faithful, so it does not make
into a concrete category.
Remark Not every category admits the structure of a concrete category. For example, Freyd showed that the category which has as objects topological spaces and as morphisms homotopy class of morphisms is not concretizable.
Consider the notion of a basis from linear algebra. We let be the forgetful functor from the category of vector spaces over
for a fixed field
to the category of sets. Let
be a vector space and let
be a subset. To phrase the notion of a basis in terms of morphisms, one can say that
is a basis for
if and only for every morphism of sets
to another vector space
, we have a unique morphism of vector spaces
such that
.
This is just a formal way of saying that a subset of a vector space is a basis if and only if one can uniquely define linear maps to other vector spaces by the values of the basis vectors.
Having phrased the notion of a basis in this way, we can immediately abstract from the special case of vector spaces to arbitrary concrete categories.
Definition C9.14 Let be a concrete category. Then for an object
, we say that a map
is a basis for
if we have the following universal property: For any object
and for any map
, there is a unique morphism
such that
. We say that an object is free if it admits a basis.
Intuitively, a basis for an object
is like a subset (as we may think of objects in a concrete category as sets by the virtue of the faithful functor that is part of the datum) except that we don’t require injectivitiy, which “generates” the object in the sense that any morphism from
to another object is determined by its restriction to
and which does so “indepedently” or “freely” that there are no “dependencies” between the elements of
in
that would restrict how we can define morphisms based on where the elements in
are sent.
Example C9.15 We already saw that bases in the sense of linear algebra are a special case of the abstract notion. The same holds for modules over a ring, although in general not every module is free.
Exercise C9.16 Consider the concrete category of topological spaces with the forgetful functor to
, then a topological space is a free object if and only if the topology is discrete, and in this case the only possible basis is the whole space.
One can easily relate the property of a basis with representable functors:
Lemma C9.17 Let be a concrete category and let
be a set. Then there exists a free object in
with basis
if and only if the functor
is representable. In this case, a representing object corresponds to a free object and the universal element corresponds to the basis.
Proof This follows directly from lemma C5.3 in A Brief Introduction to Categories, Part 5: Universal Properties and Representable Functors.
One can observe that for modules over a ring , a module is free if and only if it is isomorphic to a direct sum of copies of
and for topological spaces, a space is discrete if and only if it is homeomorphic to a disjoint union of singletons. The following lemma generalizes these obvservations.
Lemma C9.18 Let be a concrete category. Suppose that
is representable by some object
. Then an object is free if and only if it isomorphic to a coproduct of copies of
.
Proof Let be any index set and consider the coproduct
We have natural isomorphisms
. We conclude by lemma C9.17 that
is free with basis
.
Suppose that is free with basis
. Then by lemma C9.17 we have a natural isomorphism
. By the computation from the other direction, the latter Hom set is naturally isomorphic to
. We conclude that
by the Yoneda lemma.
Example C9.19 Consider the category of small categories with the concrete category structure given by the functor
that sends a small category
to the set of all morphisms in
.
is represented by the arrow category (cf. C3.6 and C5.7). Thus lemma C9.18 implies that a small category is free if and only if it is a disjoint union of copies of the arrow category. This means that for a set
, the free category generated by
is given by having for each
, two objects
and a unique morphism
(apart from the required identities).
Here’s a representation of the free category on three arrows (the identities are not depicted):
At this point, I should justify why we talk about free objects in a post about adjunctions. We will generalize the situation first.
Adjunctions via Universal Morphisms
Definition C9.20 Let be categories and let
be a functor. Then we say that for an object
, there exists a local left adjoint at
if there is an object
and a morphism
with the following universal property: For every object
and every morphism
there is a unique morphism
such that
.
Local left adjoints are like bases. Clearly we have a free object in a concrete category with basis
, if and only if
has a local left adjoint at
.
The reason for name “local left adjoint” is the following theorem:
Theorem C9.21 Let be categories and let
be a functor. Then
has a left adjoint if and only if for every object
,
has local left adjoint at
.
Proof Suppose that has a left adjoint
. Let
be an object. Then we have the unit
, so we can set
,
. Let
be an object. As we have seen in lemma C9.6, the map
is a bijection, which means precisely that for every morphism
there is a unique morphism
such that
.
Suppose that for every object ,
has local left adjoint at
. Call the local left adjoint of
at
. We need to extend the assignment
to a functor. Let
be a morphism in
. We have morphisms
and
satisfying the respective universal properties. Thus we have
which gives us a unique morphism
such that
.
We now have to check functoriality: let and suppose we are given morphisms
and
. By construction of
and
, we have
and
. We thus obtain
. But
also satisfies
and so by uniqueness, we obtain
. The fact that
preserves identities follows by a similar argument: both
and
satisfy the equation
.
We now have to check that is left adjoint to
.
Since is a local left adjoint to
for any
, we have for any morphism
a unique morphism
such that
. This implies that the map
is a bijection.
In equations, for any , we need to check that
, that is
. But this follows from the functoriality of
and the fact that
, which was already mentioned in the construction of
.
Adjunctions via Representability
Due to the close connections between representability and universal properties that were established by lemma C5.3 in A Brief Introduction to Categories, Part 5: Universal Properties and Representable Functors, we can immediately conclude from theorem C5.21:
Corollary C9.22 A functor has a local left adjoint at
if and only if the functor
is representable.
Example C9.23 Let be a category and let
be a small category, thought of as an index category. Then the diagonal functor
, which sends an object
to the constant functor with value
has a local left adjoint at an object
if and only if the colimit over
exists in
.
Proof Apply the dualized version of lemma C7.10 in A Brief Introduction to Categories, Part 7: General Limits and Colimits.
Combining theorem C9.21 and corollary C9.22, we obtain:
Corollary C9.24 A functor has a left adjoint if and only if for every
, the functor
is representable.
Dualizing, we obtain:
Corollary C9.25 A functor has a right adjoint if and only if for each
, the functor
is representable.
Example C9.26 Let be a category and let
be a small category, thought of as an index category. Then the diagonal functor
, which sends an object
to the constant functor with value
has a left adjoint if and only if for all functors
, the colimit
exists in
. Dually, this functor has a right adjoint if and only if the limits for all
exist.
Thus one can think of adjoint functors as a parametrized version of representable functors, providing yet another view on this important concept. We can also turn this around and see that representable functors are special case of adjoint functors. To see this, we use the characterization of representability via the existence of initial objects in a certain category.
Lemma C9.27 Let be a category. Then
has an initial object if and only if the terminal functor
has a left adjoint. Here
is the category with one object and one morphism. If
is a left adjoint, then
evaluted at the unique object of
is an initial object.
Proof Exercise. (There’s not much to check)
Corollary C9.28 A functor is representable if and only if the terminal functor
has a left adjoint. Here
is the category of elements, as defined here in definition C5.14. In this case, the value of a left adjoint at the unique object of
is a pair
, where
is a representing object and
is a universal element.
This follows directly from lemma C9.24 and lemma C5.15 in A Brief Introduction to Categories, Part 5: Universal Properties and Representable Functors.