Universal Properties and Adjunctions

“Free” constructions are abundant in Algebra, and are actually examples of Adjunctions. Specifically, we have an adjunction:

$$\mathbf{Set}\underset{?}{\overset{F}{\rightleftarrows }}\mathbf{Alg}$$

Where $F:\mathbf{Set}\to \mathbf{Alg}$ is the free functor from the category of sets to this category of algebraic structures, and $?:\mathbf{Alg}\to \mathbf{Set}$ is the forgetful functor, which forgets the additional algebraic structure.

Usually, this universal construction is presented a bit differently, but is equivalent to the notion of adjunction.

Free Groups

As a concrete example, take Free Groups. Given a set $A$, the free group $FA$ is usually defined with a universal property:

There is a group $FA$, and a set function $\eta :A\to FA$ such that for any other group $G$ and function $g:A\to G$, there exists a unique group homomorphism $\phi !:FA\to G$ making the following diagram commute:

Of course, when talking about the interaction of morphisms and objects from $\mathbf{Grp}$, we really mean their images, under the forgetful functor $?:\mathbf{Grp}\to \mathbf{Set}$. Being explicit, we get the following diagram:

We can characterize this object $FA$ as being an initial object in the slice category $A\downarrow ?$ (where $A:1\to \mathbf{Set}$ is the functor sending every object to $A$, and every morphism to $1_A$).

The objects consists of a choice of group $G$ and set function $A\to ?G$, and the morphisms are group homomorphisms $G\to H$ making the following diagram commute.

It’s clear that the initial object in this category is the free group $FA$, along with the morphism $\eta :A\to ?FA$.

Adjunctions

It turns out that this characterization is implied by the adjunction:

$$F⊣?$$

The “primal” characterization of an adjunction $L⊣R$ is that there exists a special binatural isomorphism between the two functors:

$$\mathcal{D}(L-,-)\cong \mathcal{C}(-,R-)$$

A more convenient (and entirely equivalent) characterization is that of two natural transformations:

$$\begin{array}{r}\eta :1\Rightarrow RL\\ ϵ:LR\Rightarrow 1\end{array}$$

Satisfying these two relations:

which we call the left and right “zig-zag identities”.

Note:

I use the notation ${\alpha }_F$ to denote the natural transformation whose component for the object $A$ is ${\alpha }_{FA}$.

Similarly, $G\beta$ denotes the natural transformation whose component at $B$ is $G{\beta }_B$.

Implying the Universal Property

With this adjunction in place, we can show that $(LA,{\eta }_A)$ is the initial object in the comma category $A\downarrow R$, for any object $A$.

This is a fun series of diagram chases.

First, we show that there is a morphism to the other objects in the comma category. Given $(M,f:A\to RM)$ some other object in $A\downarrow R$, we provide a morphism $\phi :LA\to M$ such that this diagram commutes:

Applying the right zig-zag identity at $M$, we get this commuting diagram:

We can compose this with $f$ to get:

But then, by naturality of $\eta$, we have:

And then $Lf⋙{ϵ}_M$ is our $\phi$. Since $R$ is a functor $RLf⋙R{ϵ}_M=R(Lf⋙{ϵ}_M)$

Note:

In the specific case of a free group, $Lf=Ff$ works by preserving all of the free algebraic operations we’ve created, but swapping out the “seeds” we’ve used. Then we interpret these free operations as concrete operations in $M$, using $ϵ$.

Next, we show that this function is unique. In other words, if $\varphi :LA\to M$ with ${\eta }_A⋙R\varphi =f$, then $\varphi =\phi$.

We start with:

Then, we apply the left zig-zag identity, to get:

By naturality of $ϵ$, we get:

But, since ${\eta }_A⋙R\varphi =f$, we have:

But, by definition $\phi =Lf⋙{ϵ}_M$, giving us:

In other words $\phi =\varphi$.

The Dual Construction

If we take the opposite functors $L^{op}:{\mathcal{C}}^{op}\to {\mathcal{D}}^{op}$ and $R^{op}:{\mathcal{D}}^{op}\to {\mathcal{C}}^{op}$, we have:

$$R^{op}⊣L^{op}$$

Using our previous result, we get that for any object $M$ in ${\mathcal{D}}^{op}$, $(R^{op}M,{ϵ}_M:M\leftarrow LRM)$ is initial in the slice category $M\downarrow L^{op}$

Of course, this is the same thing as saying that $(RM,{ϵ}_M)$ is terminal in the slice category $L\downarrow M$. In other words, for any object $A$ in $\mathcal{C}$, with a morphism $\alpha :LA\to M$ in $\mathcal{D}$, there exists a unique $f:A\to RM$ such that this diagram commutes:

Concretely

Back to the example of free groups, we have the following situation:

Ultimately, this just expresses the fact that a group homomorphism out of a free group consists first of replacing each of the seeds with an element of $G$, and then reducing the free algebraic structure using $ϵ$.

The Other Direction

We can also go in the other direction. Let $L:\mathcal{C}\to \mathcal{D}$, $R:\mathcal{D}\to \mathcal{C}$ be functors. If we have parameterized functions (which we don’t yet assume to be natural) ${\eta }_A:A\to RLA$ for any object $A$ in $\mathcal{C}$, and ${ϵ}_M:LRM\to M$ for any object $M$ in $\mathcal{M}$, such that $(LA,{\eta }_A)$ is initial in $A\downarrow R$, and $(RM,{ϵ}_M)$ is terminal in $L\downarrow M$, then we have an adjunction:

$$L⊣R$$

First, $\eta$ is natural:

The right and left triangles both commute, since they make use of the unique $\phi !$ that must exist whenever we have a function $A\to RM$, for some $M$.

Secondly, $ϵ$ is natural:

The left and right triangles both commute, making use of the universal property of $LRA$. (The argument is similar to before, of course).

Now, for the zig-zag identities.

Let’s start with the right zig-zag identity, for some object $M$:

If we take $1:RM\to RM$, we have a unique $\phi !$ such that:

by the universal property for $RL$.

For any $\phi :LRM\to M$, we have a unique $f!$ such that:

by the universal property for $LR$.

Combining the first diagram, with the image under $R$ of the second, we get:

But, $Lf!⋙{ϵ}_M$ satisfies the universal property of $\phi !$, which is unique. This means that $\phi =Lf!⋙{ϵ}_M$.

We then have:

But clearly, $1$ satisfies the universal property of $f!$ in this situation, which means $f!=1$. This gives us:

And so the right zig-zag property is satisfied.

For the left zig-zag property, we use the same strategy.

Given any object $A$, in $\mathcal{C}$ we want to show:

We have the following diagram:

With the unique $f!$ satisfying this diagram existing because of the universal property of $LR$.

Similarly, for any $f$, we have:

because of the universal property of $RL$.

Combining both diagrams, we get:

A similar argument as last time shows us first that $f!=\eta ⋙R\phi !$, and then $\phi !=1$, giving us:

and so the left zig-zag property is satisfied.

Conclusion

Given functors $L:\mathcal{C}\to \mathcal{D}$ and $R:\mathcal{D}\to \mathcal{C}$, these statements are equivalent:

1. $L⊣R$
2. $\mathcal{D}(L-,-)\cong \mathcal{C}(-,R-)$
3. There exist $\eta :1\Rightarrow RL,ϵ:LR\Rightarrow 1$ satisfying the zig-zag identities
4. ($\forall A\in \mathcal{C},M\in \mathcal{D}$) $(RLA,{\eta }_A)$ is initial in $A\downarrow R$, and $(LRM,{ϵ}_M)$ is terminal in $L\downarrow M$.

In this post, I only proved $3⟺4$. $1⟺2$ by definition, and $2⟺3$ is well known.