The Unapologetic Mathematician

Mathematics for the interested outsider

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:
\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