Today I return to the discussion of universals, limits, representability, and related topics. The last piece of this puzzle is the notion of an adjunction. I’ll give a definition and examples today and work out properties later.
An adjunction between categories and consists of a pair of functors and and a natural isomorphism . Notice that the functors on either side of go from to , so each component is a bijection of sets. We say that is “left adjoint” to , and conversely that is “right adjoint” to , and we write .
Now, we have been seeing these things all along our trip so far, but without mentioning them as such. For instance, we have all the “free” constructions:
- the free monoid on a set
- the free group on a set
- the free group on a monoid
- the semigroup ring
- the free ring on an abelian group
- the free module on a set
- the free algebra on a module
and maybe more that I’ve mentioned, but don’t recall.
These all have a very similar form in their definitions. For instance, the free monoid on a set is characterized by the following universal property: every function from into the underlying set of a monoid extends uniquely to a monoid homomorphism . If we write the underlying set of as , we easily see that is a functor. The condition then is that every element of the hom-set corresponds to exactly one element of the hom-set , and every monoid homomorphism restricts to a function on . That is, for every set and monoid we have an isomorphism of sets .
Now, given a function from a set to a set we can consider to be a subset of the free monoid on itself, giving a function . This extends to a unique monoid homomorphism . This construction preserves identities and compositions, making into a functor from to .
If we have a function and a monoid homomorphism then we can build functions and . The isomorphisms and commute with these arrows, so they form the components of a natural isomorphism between the two functors. This proves that the free monoid functor is a left adjoint to the forgetful functor .
All the other examples listed above go exactly the same way, giving left adjoints to all the forgetful functors.
As a slightly different example, we have a forgetful functor that takes an abelian group and “forgets” that it’s abelian, leaving just a group. Conversely, we can take any group and take the quotient by its commutator subgroup to get an abelian group. This satisfies the property that for any group homomorphism from to an abelian group (considered as just a group) there is a unique homomorphism of abelian groups . Thus it turns out that “abelianization” of a group is left adjoint to the forgetful functor from abelian groups to groups.
There are more explicit examples we’ve seen, but I’ll leave them to illustrate some particular properties of adjoints. Take note, though, that not all adjunctions involve forgetful functors like these examples have.
An adjunction between two categories can be seen as a weaker version of an equivalence. An equivalence given by functors and tells us that both and are fully faithful, so . Now let’s put to find that , where the last isomorphism uses the natural isomorphism . So every equivalence is an adjunction.