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.
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:
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 .
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 any , 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.