# The Unapologetic Mathematician

## Mathematics for the interested outsider

Today I return to the discussion of universals, limits, representability, and related topics. The last piece of this puzzle is the notion of an adjunction. I’ll give a definition and examples today and work out properties later.

An adjunction between categories $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$ and a natural isomorphism $\Phi_{X,Y}:\hom_\mathcal{D}(F(X),Y)\rightarrow\hom_\mathcal{C}(X,G(Y))$. Notice that the functors on either side of $\Phi$ go from $\mathcal{C}^\mathrm{op}\times\mathcal{D}$ to $\mathbf{Set}$, so each component $\Phi_{X,Y}$ is a bijection of sets. We say that $F$ is “left adjoint” to $G$, and conversely that $G$ is “right adjoint” to $F$, and we write $F\dashv G$.

Now, we have been seeing these things all along our trip so far, but without mentioning them as such. For instance, we have all the “free” constructions:

and maybe more that I’ve mentioned, but don’t recall.

These all have a very similar form in their definitions. For instance, the free monoid $M(S)$ on a set $S$ is characterized by the following universal property: every function $f$ from $S$ into the underlying set of a monoid $N$ extends uniquely to a monoid homomorphism $\bar{f}:M(S)\rightarrow N$. If we write the underlying set of $N$ as $U(N)$, we easily see that $U:\mathbf{Mon}\rightarrow\mathbf{Set}$ is a functor. The condition then is that every element of the hom-set $\hom_\mathbf{Set}(S,U(N))$ corresponds to exactly one element of the hom-set $\hom_\mathbf{Mon}(M(S),N)$, and every monoid homomorphism restricts to a function on $S$. That is, for every set $S$ and monoid $N$ we have an isomorphism of sets $\hom_\mathbf{Mon}(M(S),N)\cong\hom_\mathbf{Set}(S,U(N))$.

Now, given a function from a set $S_1$ to a set $S_2$ we can consider $S_2$ to be a subset of the free monoid on itself, giving a function $f:S_1\rightarrow U(M(S_2))$. This extends to a unique monoid homomorphism $M(f):M(S_1)\rightarrow M(S_2)$. This construction preserves identities and compositions, making $M$ into a functor from $\mathbf{Set}$ to $\mathbf{Mon}$.

If we have a function $f:S_1\rightarrow S_2$ and a monoid homomorphism $m:N_1\rightarrow N_2$ then we can build functions $\hom_\mathbf{Mon}(M(f),m):\hom_\mathbf{Mon}(M(S_2),N_1)\rightarrow\hom_\mathbf{Mon}(M(S_1),N_2)$ and $\hom_\mathbf{Set}(f,U(m)):\hom_\mathbf{Set}(S_2,U(N_1))\rightarrow\hom_\mathbf{Set}(S_1,U(N_2))$. The isomorphisms $\hom_\mathbf{Mon}(M(S_2),N_1)\cong\hom_\mathbf{Set}(S_2,U(N_1))$ and $\hom_\mathbf{Mon}(M(S_1),N_2)\cong\hom_\mathbf{Set}(S_1,U(N_2))$ commute with these arrows, so they form the components of a natural isomorphism between the two functors. This proves that the free monoid functor $M:\mathbf{Set}\rightarrow\mathbf{Mon}$ is a left adjoint to the forgetful functor $U:\mathbf{Mon}\rightarrow\mathbf{Set}$.

All the other examples listed above go exactly the same way, giving left adjoints to all the forgetful functors.

As a slightly different example, we have a forgetful functor $U:\mathbf{Ab}\rightarrow\mathbf{Grp}$ that takes an abelian group and “forgets” that it’s abelian, leaving just a group. Conversely, we can take any group $G$ and take the quotient by its commutator subgroup $\left[G,G\right]$ to get an abelian group. This satisfies the property that for any group homomorphism $f:G\rightarrow U(A)$ from $G$ to an abelian group $A$ (considered as just a group) there is a unique homomorphism of abelian groups $\bar{f}:G/[G,G]\rightarrow A$. Thus it turns out that “abelianization” of a group is left adjoint to the forgetful functor from abelian groups to groups.

There are more explicit examples we’ve seen, but I’ll leave them to illustrate some particular properties of adjoints. Take note, though, that not all adjunctions involve forgetful functors like these examples have.

An adjunction between two categories can be seen as a weaker version of an equivalence. An equivalence given by functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$ tells us that both $F$ and $G$ are fully faithful, so $\hom_\mathcal{C}(C',C)\cong\hom_\mathcal{D}(F(C'),F(C))$. Now let’s put $C'=G(D)$ to find that $\hom_\mathcal{C}(G(D),C)\cong\hom_\mathcal{D}(F(G(D)),F(C))\cong\hom_\mathcal{D}(D,F(C))$, where the last isomorphism uses the natural isomorphism $F\circ G\rightarrow1_\mathcal{D}$. So every equivalence is an adjunction.

July 16, 2007 - Posted by | Category theory

1. A very nice introduction to adjoints. I found them scary when I didn’t know them but now one of my favorite things in math is to “play around with adjunctions”, which Grothendieck called ‘functor yoga’!

2. […] Unit and Counit of an Adjunction Let’s say we have an adjunction . That is, functors and and a natural isomorphism […]

Pingback by The Unit and Counit of an Adjunction « The Unapologetic Mathematician | July 17, 2007 | Reply

3. You might enjoy a physicist applying functor theory to elementary particles, kea-monad.blogspot.com. It’s all quite enough to make my head hurt.

Comment by carlbrannen | July 19, 2007 | Reply

4. […] and universality Now we have the notion of an adjunction, along with its unit and counit. It’s time to start tying them back into […]

Pingback by Adjoints and universality « The Unapologetic Mathematician | July 19, 2007 | Reply

5. […] of Adjoints And now we go back to adjoints. Like every other structure out there, we want to come up with some analogue of a homomorphism […]

Pingback by Transformations of Adjoints « The Unapologetic Mathematician | July 30, 2007 | Reply

6. […] gives us a “forgetful” functor which returns the underlying sets and functions. Then as we saw, we often have a left adjoint to this forgetful functor giving the “free” structure […]

Pingback by Free Monoid Objects « The Unapologetic Mathematician | August 2, 2007 | Reply

7. […] they give another 2-category: . The objects here again are categories, but now the 1-morphisms are adjunctions between categories. And then we have conjugate pairs between adjunctions, with the […]

Pingback by 2-Categories « The Unapologetic Mathematician | August 16, 2007 | Reply

8. […] But I’ve been thinking about it. I’ll also have to check up on some of my references to see if they can tell me how to do what I’m pretty sure can be done, or I’ll just have to cobble it together myself. But it’s pretty cool. I’ll give you one hint: look how I started my discussions of adjoint functors. […]

Pingback by Nothing today « The Unapologetic Mathematician | August 25, 2007 | Reply

9. […] When I started in on adjoint functors, I gave the definition in terms of a bijection of hom-sets. Then I showed that we can also specify […]

Pingback by Enriched Adjunctions « The Unapologetic Mathematician | September 4, 2007 | Reply

10. I agree with ulfarsson that this is a very nice presentation. I was surprised when I first tried to learn about these things that the discussion in Mac Lane (1971), which, like this one, focusses on the bijection between Hom-sets, made more sense to me than the more commonly found introductory discussions starting with the unit (most of Mac Lane is of course much too hard for me, which is as it should be!)

Intuitively for me, I find that the unit-based version looks like something that just comes in thru the window and lands on your plate in front of you, rather than appearing to have a sensible antecedent background (until it’s derived from the bijection, of course).

Comment by MathOutsider | October 26, 2007 | Reply

11. […] I want to mention a topic I thought I’d hit back when we talked about adjoint functors. We know that every poset is a category, with the elements as objects and a single arrow from to […]

Pingback by Galois Connections « The Unapologetic Mathematician | May 18, 2009 | Reply

12. […] should recall the relation between two adjoint functors. An important difference here is that there is no distinction between left- and right-adjoint […]

Pingback by Adjoint Transformations « The Unapologetic Mathematician | May 22, 2009 | Reply

13. […] proof is perhaps best thought of as a general phenomenon involving adjoint functors, but there is a quick (if less enlightening) direct proof […]

Pingback by Completions of rings and modules « Delta Epsilons | August 25, 2009 | Reply

14. […] algebra on the vector space . That’s a tip-off that we’re thinking should be the left adjoint of the “forgetful” functor which sends a graded-commutative algebra to its underlying […]

Pingback by Functoriality of Tensor Algebras « The Unapologetic Mathematician | October 28, 2009 | Reply

15. […] adjoint functor. For more about adjoints, see (in no particular order) posts at Concrete Nonsense, the Unapologetic Mathematician, and Topological […]

Pingback by Some adjoint functors « Annoying Precision | October 29, 2009 | Reply

16. […] we come to the real version of Frobenius reciprocity. It takes the form of an adjunction between the functors of induction and […]

Pingback by (Real) Frobenius Reciprocity « The Unapologetic Mathematician | December 3, 2010 | Reply