So last post I gave a (hurried) description of why adjoint pairs of functors are linked to this notion of “similar structure” between two categories.  In this post, I want to relate adjunctions to universal properties, and ultimately why we like adjoint pairs so much.

Say we’re working with the “free group on a set” functor, $F: \textbf{Set} \to \textbf{Grp}$.  We know that the free group satisfies a really nice universal property: Given any (set) function $f: X \to G$, where $G$ is a group, there exists a unique group homomorphism $\overline{f} : F(X) \to G$ extending $f$.   If you recall the notion of the “forgetful” functor $U: \textbf{Grp} \to \textbf{Set}$ that takes a group and gives its underlying set, the universal property of the free group on $X$ states that there is a bijection $\text{Hom}_{\textbf{Grp}}(F(X),G) \cong \text{Hom}_{\textbf{Set}}(X, U(G))$

i.e. $F \dashv U$ is an adjoint pair (of course, one should check the naturality of this bijection).  Hence the universal property here is really just this adjunction in hiding.  This idea is generalized often by saying that a category C has “free objects” if a suitably defined forgetful functor from C possesses a left adjoint.

It’s for this reason that I like to think of a pair of adjoint functors $F \dashv G$ as a sort of “globally defined” universal property, as expressed by the naturality of the bijection between hom-sets.  Indeed, we have that a functor $G: \textbf{D} \to \textbf{C}$ has a left adjoint provided that we can find, for each object $A$ of an object “ $FA$” of D and a morphism $\eta_A : A \to G(FA")$ that is universal among morphisms form $A$ to the image of $G$, i.e. for all $f: A \to GB$, there is a unique $\overline{f} : FA" \to B$ satisfying $f = G(\overline{f}) \circ \eta_A$.  I use the quotations here only to emphasize the fact that we really want the object “ $FA$” to be “the component of some suitable functor $F$ at $A$.”

I don’t know about you, but this sound a lot like: “For any functor $S: \textbf{1} \to \textbf{C}$, the comma category $(S(*) \downarrow G)$ has an initial object.” (Remember, a functor $S: \textbf{1} \to \textbf{C}$ simply selects an element of C.)  That is, this sounds A LOT like how we initially (haha, pun) constructed universal properties from comma categories.  The statement of the above paragraph then says that if we can find such an initial object for any choice of functor $S: \textbf{1} \to \textbf{C}$, then $G$ has a left adjoint.  Pretty neat, right?

## Isomorphism, Equivalence, and Adjunction

Isomorphism

As mathematicians are wont to do, whenever we have a collection of algebraic “objects,” we want to know how to “relate” them.  In the case of categories, we saw earlier that maps called functors are what we want to examine.

The next step after defining these structure preserving maps is to wonder what it means for two objects to be “essentially the same.”  In Set, bijections do this.  In Grp, group isomorphisms do this.  In Top, homeomorphisms do this.  Etc, etc.  If we were to try an analogous procedure for categories, we would say two categories C and D are isomorphic if there exist two functors: $F: \textbf{C} \to \textbf{D}$ and $G: \textbf{D} \to \textbf{C}$ such that $F \circ G = id_{\textbf{D}}$ and $G \circ F = id_{\textbf{C}}$.  Saying that two categories are isomorphic means that they are, for all intents and purposes, the same (maybe they differ in notation or something).

Equivalence

As it happens, this tends to be too restrictive a condition (i.e. categories that behave more or less the same tend to not actually be isomorphic).  What if, instead of requiring that $FG = id_\textbf{D}$ and $GF = id_\textbf{C}$, we require that these functors are naturally isomorphic to the appropriate identities?  We would then say that C and  are “equivalent” categories.

The first time I ever saw this phenomenon was in algebraic geometry, where one sees that the category of finitely generated reduced $k$-algebras is equivalent to the category of (affine) algebraic varieties over $k$ (here $k$ is a field).  Later on we saw that the category of commutative rings with unity is equivalent to the category of affine schemes equipped with their structure sheaves.  Another cool example is that the category of quasi-coherent sheaves over an affine scheme $\text{Spec}(R)$ is equivalent to the category of $R$-modules.

For a simpler example, take any poset $(X,\leq)$ and consider it as a category.  Then reversing the direction of the arrows gives an equivalent category $(X, \geq).$  Obviously, this works for any category and its opposite category.  I just like the poset case because one can visualize it quite easily.

Simply put: adjunctions are ubiquitous.  It took me a long time to see that, and I’m still wading through the ramifications.  I gave a (brief) blurb about them in the last post, but let’s up the scrutiny.  We say $F: \textbf{C} \to \textbf{D}$ and $G: \textbf{D} \to \textbf{C}$ are an adjoint pair (written $F \dashv G$) if there is a natural bijection between maps $f: A \to GB$ in C and $\overline{f}: FA \to B$ in D.
Note that, in D, we have the map $id_{FA} : FA \to FA$.  The adjunction gives a unique map $\eta_A: A \to GFA$, and likewise we have a unique map $id_{GB} : GB \to GB$ yielding $\epsilon_B : FGB \to B$.  These maps are called the “unit” and “counit” of the adjuction at $A$ or $B$.  In fact, the adjunction yields a pair of natural transformations $\eta: id_\textbf{C} \to GF$ and $\epsilon: FG \to id_\textbf{D}$.
That’s pretty neat.  It explicitly shows the “descending chain of equivalence” from isomorphism of categories, equivalence of categories, and adjunction of functors between categories.  The naturality of the unit and counit transformations from an adjunction $F \dashv G$ actually implies the “bijection” criterion, so we can really just take the unit-counit thing as a starting point.