Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better.
One way to define a subgroup of a group is as the image of a homomorphism into . Given the inclusion map , the functor in the category of groups acts contravariantly to give a map called restriction. More concretely, the restricted representation of a representation is defined simply by . Hence there is a functorial way to pass from a representation of a group to one of a subgroup .
It is not obvious, however, whether there is a functorial way to pass from a representation of back to one of . There is such a construction, which goes by the name of induction, and we will need it later. Today we’ll discuss the general category-theoretic context in which induction is understood, where it is called an adjoint functor. For more about adjoints, see (in no particular order) posts at Concrete Nonsense, the Unapologetic Mathematician, and Topological Musings.
Adjoints are the next best thing to inverses
Are putting a ball into a bag and taking a ball out of a bag inverse operations? It might seem so at first glance. Consider, however, that starting with a bag with balls in it, there are ways to take a ball out and then to put another one in, whereas there are ways to put a ball in and then take a ball out. So from a combinatorial perspective, the two are not inverses. What they are, however, is in a certain precise sense adjoints (or transposes) of each other.
Here’s one way to make that statement precise. Suppose we are given a collection of bags such that there are bags with balls. Since bags are unordered, we should associate this collection with its exponential generating function . After taking a ball out of each bag, a bag that had balls in it will now have balls in it and be in one of states depending on which ball was taken out, so we pass to the derivative
When putting a ball back in, we simply add a ball to each bag, so we multiply by to get
These operations do not commute, so they cannot be inverses of one another. However, has the following inner product: the powers of the derivative span the dual space, and the dual basis associated to the basis is precisely the basis . Sending to and extending by multiplicativity gives the inner product
and in this inner product we have the adjoint relation
Note that we are regarding simultaneously as an element of and as a multiplication operator. The algebra spanned by , which is a subalgebra of the full endomorphism algebra of , is the Weyl algebra in one variable. This will become relevant later.
If you don’t like that example, then let be the adjacency matrix of a graph. Then is the adjacency matrix of the opposite graph, where every edge is reversed. (, on the other hand, almost never has positive integer coefficients.)
There are many common examples of pairs of constructions that take algebraic objects to other algebraic objects which are not inverses, but which feel inverse-like. For example:
- Let be a perfect field and let be a Galois extension with Galois group . There is a map from intermediate fields to the subgroup of and another map from subgroups to their fixed fields . The fundamental theorem of Galois theory asserts that these maps are not inverses unless is Galois and is normal, respectively.
- Let be an algebraically closed field. There is a map from subsets of to the set of points in such that every element of the susbset vanishes and another map from subsets of to the set of polynomials that vanish on that subset. The strong Nullstellensatz asserts that these maps are not inverses unless is a radical ideal and is an affine variety, respectively.
- Let be a group. There is a map sending to its abelianization . We should interpret this map as a functor from groups to abelian groups to figure out what to pair it with: the inclusion functor . These maps are not inverses unless is abelian.
- Let be a group. There is a map sending a group to its underlying set and another map sending a set to the free group on it. These maps are never (naturally) inverses.
The first two examples are two of my favorites, so we’ll talk about them first. In the Galois case, the set of intermediate fields has a natural poset structure by inclusion, and so does the set of subgroups of . These poset structures, however, are “opposite” when related by the functions defined above – if the subgroup is large, then its fixed field is small, and if is small, then is large. How can we make this precise?
Well, we want to examine to what extent these functions are inverses of each other, so let’s compose them. It’s not hard to see that is at least as large as , and it’s also not hard to see that is at least as large as . In other words, and . Let’s define the following poset structures on the subgroups of and the intermediate fields ; we’ll say that if is a subgroup of and we’ll say that if is a subfield of . Call these posets and respectively. Now, we know that these poset structures can be thought of as categories where is a single arrow if and no arrow otherwise, and recall that for a category the category is obtained by reversing all arrows. Now I claim that both of the above observations are subsumed in the single observation that
When we recover the first observation and when we recover the second. The above may look intimidating, but when unpacked it says something very trivial: fixes every element of if and only if every element of is fixed by .
Now I claim that the functions and are order-preserving functions between and , and the order-preserving functions between posets are functors. Category theorists say that these two maps are an adjoint pair (with the equality replaced by a natural isomorphism) relating and . is the left adjoint and is the right adjoint. A pair of adjoint functors between posets is called a monotone Galois connection, and this example is the motivation for that terminology.
But we didn’t relate our posets by an adjunction – we related one poset to the opposite of the other. This relationship is called an antitone Galois connection because the maps involved are order-reversing (read: contravariant). And you can check that for ideals and varieties the exact same relationship holds.
The motivation for the term “adjoint” is that the relationship looks an awful lot like the relationship between a pair of adjoint linear transformations. In fact, as we’ve seen, in at least one situation it makes sense to think of as a categorification of the inner product. But there’s much deeper stuff going on here: Todd Trimble‘s post on adjoints discusses the relationship between adjoints and the Yoneda lemma, but I haven’t really digested this point yet.
A standard set of examples of adjoint functors are free and forgetful functors. Roughly speaking, if a category consists of objects in a category “with extra structure,” then there is a functor which “forgets” this structure. In nice cases this functor has a left adjoint which is the “free -object” associated to a -object. In typical examples we’ll take and to be an “algebra” (in the sense of universal algebra), i.e. a set equipped with operations satisfying some axioms.
As mentioned above, the standard example here is when . The free group on a set is typically defined as the group generated by the elements of with no extra relations besides those imposed by the group axioms. More precisely if then is the set of words on the alphabet with group operation concatenation and the relations imposed by the group axioms. Let denote the underlying set of the group , and let’s verify the adjoint relation
But this is straightforward. A function is just a function which assigns a group element to every element . And the free group has the property that, since is generated by the elements , replacing them with elements of another group gives a valid group homomorphism sending to as expected, and moreover every group homomorphism arises in this way. This is what it really means to be a free group: homomorphisms out of a free object should be as unconstrained as possible, which is essentially the universal property of the free group construction.
Another nice example is when (where we fix the base field ). One thinks of an algebra as a vector space equipped with a bilinear map satisfying associativity. The forgetful functor just ignores this map, so what does its adjoint look like? Well, if we want to imitate the above construction, then what we should do is to define formal multiplication on subject to no constraint other than associativity. This will be easiest to do given a basis . First we need to introduce an identity . To define a product of two vectors we’ll introduce formal symbols and require bilinearity. This defines the tensor product , which is universal for bilinear maps . But since we also want to define products of three or more vectors we need to introduce more formal symbols , which is done by considering the higher tensor powers . The tensor product is not, strictly speaking, associative, but there is a natural association given by various maps such as . Since there are no relations imposed between these vectors, the free algebra is a direct sum of each of these tensor powers, i.e.
where . This defines the tensor algebra on , and it has the following universal property: any linear map where is an algebra extends to a unique algebra homomorphism in the obvious way. In other words, we have the adjoint relation
In the next post we’ll apply these ideas to constructing the induced representation.