# The Unapologetic Mathematician

## The Unit and Counit of an Adjunction

Let’s say we have an adjunction $F\dashv G:\mathcal{C}\rightarrow\mathcal{D}$. That is, functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$ and a natural isomorphism $\Phi_{C,D}:\hom_\mathcal{D}(F(C),D)\rightarrow\hom_\mathcal{C}(C,G(D))$.

Last time I drew an analogy between equivalences and adjunctions. In the case of an equivalence, we have natural isomorphisms $\eta:1_\mathcal{C}\rightarrow G\circ F$ and $\epsilon:F\circ G\rightarrow1_\mathcal{D}$. This presentation seems oddly asymmetric, and now we’ll see why by moving these structures to the case of an adjunction.

So let’s set $D=F(C')$ like we did to show that an equivalence is an adjunction. The natural isomorphism is now $\Phi_{C,F(C')}:\hom_\mathcal{D}(F(C),F(C'))\rightarrow\hom_\mathcal{C}(C,G(F(C'))$. Now usually this doesn’t give us much, but there’s one of these hom-sets that we know has a morphism in it: if $C'=C$ then $1_{F(C)}\in\hom_\mathcal{D}(F(C),F(C))$. Then $\Phi_{C,F(C)}(1_{F(C)})$ is an arrow in $\mathcal{C}$ from $C$ to $\left[G\circ F\right](C)$.

We’ll call this arrow $\eta_C$. Doing this for every object $C\in\mathcal{C}$ gives us all the components of a natural transformation $\eta:1_\mathcal{C}\rightarrow G\circ F$. For this, we need to show the naturality condition $G(F(f))\circ\eta_C=\eta_{C'}\circ f$ for each arrow $f:C\rightarrow C'$. This is a straightforward calculation:
$G(F(f))\circ\eta_C=G(F(f))\circ\Phi_{C,F(C)}(1_{F(C)})=\Phi_{C,F(C')}(F(f)\circ1_{F(C)})=$
$\Phi_{C,F(C')}(1_{F(C')}\circ F(f))=\Phi_{C',F(C')}(1_{F(C')})\circ f=\eta_{C'}\circ f$
using the definition of $\eta_C$ and the naturality of $\Phi$.

This natural isomorphism $\eta$ is called the “unit” of the adjunction $F\dashv G$. Dually we can set $C=G(D)$ and extract an arrow $\eta_D=\Phi_{G(D),D}^{-1}(1_{G(D)})$ for each object $D\in\mathcal{D}$ and assemble them into a natural transformation $\eta:F\circ G\rightarrow1_\mathcal{D}$ called the “counit”. If both of these natural transformations are natural isomorphisms, then we have an equivalence.

For a particular example, let’s look at this in the case of the free-monoid functor $M$ as the left adjoint to the underlying-set monoid $U$. The unit will give an arrow $\eta_S:S\rightarrow U(M(S))$, which here is just the inclusion of the generators (elements of $S$) as elements of the underlying set of the free monoid. The counit, on the other hand, will give an arrow $\epsilon_N:M(U(N))\rightarrow N$. That is, we take all elements of the monoid $N$ and use them as generators of a new free monoid — write out “words” where each “letter” is a whole element of $N$. Then to take such a word and send it to an element of $N$, we just take all the letters and multiply them together as elements of $N$. Since we gave a description of $\Phi$ last time for this case, it’s instructive to sit down and work through the definitions of $\eta_S=\Phi_{S,M(S)}$ and $\epsilon=\Phi_{U(N),N}^{-1}$ to show that they do indeed give these arrows.

July 17, 2007 Posted by | Category theory | 11 Comments