2024-10-08

This following is a very brief summary of what happened in class on 2024-10-08.

We introduced three relationships that might exist between a pair of functors $F$ and $G$ (going opposite directions between two categories $C$ and $D$). We motivated our ideas by considering the situation of objects $c\in C$ and $d\in D$ and wondering how to consider arrows "between them". With the functors $F$ and $G$ at our disposal, we had two options:

1. We could use $F$ to send $c$ into $D$, and then consider all arrows $F(c)\to d$ in $D$
2. We could use $G$ to send $d$ into $C$, and then consider all arrows $c\to G(d)$ in $C$

In other words, we could consider the two sets ${\mathrm{Hom}}_C(F(c),D)$ and ${\mathrm{Hom}}_C(c,G(d))$. For a random pair of functors, there is usually no relationship between these two sets. But what if the functors $F$ and $G$ weren't random? We considered some situations:

1. Case 1: $F$ and $G$ are mutual inverses
   In this case we have $GF=I_C$ and $FG=I_D$, and both functors are said to be isomorphisms. We then say that the categories $C$ and $D$ are isomorphic. The functors $F$ and $G$ establish bijections both between the objects of each category, and the arrows of each category. The two categories are, for all intents and purposes, essentially the same. So here we definitely have a (somewhat trivial) bijection between the hom-sets considered above.

2. Case 2: $F$ and $G$ are "almost" inverses
   In this case we are suppose $GF\simeq I_C$ and $FG\simeq I_D$, i.e., the two compositions are natural isomorphic to the identity functors. This is more flexible than the notion of isomorphism, and the corresponding adjective for the categories $C$ and $D$ is to say they are equivalent. It turns out that this is essentially the same as saying that the two categories are "the same up to isomorphism classes of objects." In other words, there might not be a bijection on objects, but if we identified isomorphic objects in each category, the results would look the same.

3. Case 3: $F$ and $G$ are "related"
   In this case we supposed that there was a natural bijection between the hom-sets we originally considered. In other words, for every $c\in C$ and $d\in D$ there is a bijection

   $${\tau }_{c,d}:{\mathrm{Hom}}_D(F(c),d)\stackrel{\sim }{\to }{\mathrm{Hom}}_D(c,G(d)).$$

   We'll make the "naturality" precise next class.

We focused our attention on the last case, in which case we say the two functors are adjoints. We also call the functor $F$ the left adjoint (because in the bijection it appears in the left component of the hom-set), while the functor $G$ is a right adjoint.

We noted that the natural bijection $\tau$ can also be used to construct natural transformations $\eta :I_C\Rightarrow GF$ and $\epsilon :FG\Rightarrow I_D$, called the unit and counit of the adjunction, respectively. These are the relaxations of the natural isomorphisms of Case 2, above.

We then briefly talked about an example, namely the free abelian group functor as being left adjoint to the corresponding forgetful functor.

We will talk more about adjoints next class!

Concepts

• Adjoints

References

• Mac Lane: pages 79-89