# Adjoints

...good general theory does not search for the maximum generality, but for the right generality.

# Motivation

Have you ever wondered what's so "free" about free groups? Or why the free abelian group on a set

Or perhaps you've wondered if objects in different categories can have maps to each other. Sadly, they cannot. But they can try the next best thing. Imagine for a moment, two categories

If there are no functors between

But suppose there *were* functors connecting the two categories. We can think of these as transit lines, able to convey data (objects and arrows) from one category to the other.

Suddenly it *is* possible for the two objects to interact. We just need to send one of the objects into the other category. Of course, we have two options. We can either send

And now we have two different sets of maps we can consider. We can consider the maps between

For a random pair of functors

## Special Case 1: Isomorphisms of categories

Let's first consider the case in which **isomorphisms** and the categories

Is this case interesting? Not really. An isomorphism of categories means a bijection on both objects and arrows. For all intents and purposes, the categories

## Special Case 2: Equivalences of categories

With natural transformations at our disposal, it makes sense to relax the requirement that the compositions *naturally isomorphic* to the respective identity functors. In other words, let's assume there are natural isomorphisms **naturally isomorphic** and the categories **equivalent**.

When people talk about equivalent categories, this is what they mean. For our purposes, though, this is still too strong of a condition on our functors. Equivalent categories are "basically" the same, and we want to allow for functors between categories that are "very different."

## Special Case 3: Adjunction of categories

Suppose we stick to our original idea, and ask for pairs of functors

Inverse functors and isomorphic functors certainly satisfy this condition, but potentially many other pairs of functors, too. Following this thread leads to the notion of an **adjunction** of categories.

# Adjunction of categories

There a several equivalent definitions, but we'll give the one that most resembles the situations we've seen:

Suppose **adjunction** from

and

We call the pair of functors **adjoint pair**. The functor **left adjoint** for **right adjoint** for

We should immediately note that the adjunction requires all three pieces of information: the functor

## The naturality condition

The statement that

Similarly, naturality in

In short, the bijections in

The description of an adjunction can be rephrased in an equivalent way in which

We will see a *plethora* of examples (below), but first a few additional notes are in order.

# Units and counits

There is a lot of additional information that can be extracted from an adjunction, and some of that information is equivalent to the adjunction itself. Here we list some of that data.

If we set

We call **unit** of the adjunction. This situation is visualized below:

This unit

The image above might suggest that

Similarly, if we set

We call **counit** of the adjunction. This dual situation is visualized below:

Moreover, the unit and counits satisfy two "triangular" identities, namely the commutative triangles of natural transformations below:

What should you take away from these triangles? Not much. Only that these identities give specific natural isomorphisms of

In any case, it turns out that an adjunction between