Making sense of adjunctions

When I first encountered the concept of an adjunction, I got quite confused as to what it is, and why it is useful: Just how on earth is a left adjoint of a functor? What’s the matter of a free and forgetful functor, why is free left to forgetful but not the other way round. That’s where this blog comes from. I hope this blogpost can help demystify adjunction for you a little bit.

Disclaimer: I am still not very certain that I understand this concept very well so there might be mistakes/misunderstandings in this post. This is merely an attempt to understand adjunction more intuitively through many examples. Please read it with a grain of salt, and, as always, let me know if you spot a mistake.

Definition

Let’s first define what an adjunction is. It turns out there are many equivalent definitions, I will be using this one¹:

An adjunction between two categories \(C\) and \(D\) is specified by:

• functors \(F\) and \(G\)

• For each \(X\in C\) and \(Y \in D\) a bijection \(\theta_{X,Y} \): \(D(F X, Y) \cong C(X,G Y) \) which is natural in \(X\) and \(Y\).

Roughly speaking, this means if \(F \dashv G\) there is a morphism in \(D: F X \to Y\), then we can find a morphism in \(C: X\to G Y\).

Examples of adjunctions

We list a few examples of adjunctions:

Free and forgetful

The free functor is left adjoint to the forgetful functor \(F \dashv U\). In this case the free functor takes a set and converts it into a list monoid whose list elements come from the set, and with the list concatenation and empty list as the monoid operation and identity element.

$$ \begin{prooftree} \AxiomC{\(\Sigma \to U(M,\cdot, e)\)} \UnaryInfC{\(F\ \Sigma \to (M,\cdot, e)\)} \end{prooftree} $$

Due to the universal properties that when we can go from a set to (the set component of) a monoid, we can always apply that function to every element in the list and hence go from the list monoid to the monoid.

Diagonal and product and coproduct

The diagonal functor is left adjoint to the product functor \(\Delta \dashv \times\)

$$ \begin{prooftree} \AxiomC{\(\Delta C=(C,C) \to (X,Y)\)} \UnaryInfC{\(C \to \times(X, Y)=X\times Y\)} \end{prooftree} $$

This is due to the terminal nature of a binary product in a category. We see that we have a morphism to go from \(C\) to \(X\times Y\), then this implies we can go from \(C\) to \(X\) and \(Y\), which means there is indeed a unique morphism from \(C\) to the binary product (by the UP of binary product).

Similarly, the coproduct is actually left adjoint to the diagonal functor \(+ \dashv \Delta \dashv \times \).

Initial and terminal object

Suppose \(!: \mathbf C\to \mathbf 1\) is a functor that maps an arbitrary category to the terminal category, then it is left adjoint to the functor that maps the unique object in the terminal category the terminal object to any other category \(C\).

$$ \begin{prooftree} \AxiomC{\(! C\to *\)} \UnaryInfC{\(C \to U(*)\)} \end{prooftree} $$

Why is that the case? Looking at the required property: for each \(\overline f\), we need to have a unique \(f\) that maps from any object \(C\) to the object \(U(*)\). \(f\) always exists since there is only one object in the terminal category, hence we need to have an always-existing \(\overline f\) as well, which forces the \(U(*)\) to be terminal.

Similarly \(F\dashv\ ! \dashv U\) where \(F\) maps an object \(*\) to the initial object.

Thoughts

So what is the pattern here? The more left a functor is, the more initial it will be and dually, the more right a functor is, the more terminal its destination object would be.

Looking at the diagram, we note that for each \(f\), there is a unique \(\overline f\) such that the triangle commutes, and vice versa. If we were to fix \(R\) and try to find \(L\), then we would want \(L X\) to be as “initial” as possible, since an initial object, by definition, has a unique morphism to any other object in the category. Dually, if we fix \(L\) and want to find \(R\), then we would want \(R X\) to be as terminal as possible, since we want to go from \(X\) to \(R Y\).

References

Awodey, S. (2010). Category theory. Oxford University Press (2nd ed.).