# Adjunctions in the real world

What concepts in the real world can be described by adjunctions?

For example, parents and children are adjoint to one another. Specifically, work in $$ZFC$$ plus a finite class of atoms $$\mathscr{X}$$ (one for each person) and let $${\sf Par}:\mathcal{P}(\mathscr{X})\to\mathcal{P}(\mathscr{X})$$ $${\sf Chi}:\mathcal{P}(\mathscr{X})\to\mathcal{P}(\mathscr{X})$$ the functions sending a collection of atoms to the collection of all the atoms corresponding to their parents or children, respectively. Then $${\sf Par\dashv Chi}$$ since we have a (covariant) Galois connection between the poset $$\big(\mathcal{P}(\mathscr{X}),\subseteq\big)$$ and itself given by the above functions (your parent's children includes you and your children's parents also includes you*), and a covariant Galois connection is an adjunction. What are some other examples?

*Technically, those of us without children (myself currently included) form a hole in the argument as it stands -- we get sent to the empty set by $${\sf Chi}$$, then back to the empty set by $${\sf Par}$$ again instead of a set of people containing ourselves. To fix this, add an additional atom $${\sf f}$$ to the theory called free time, and let $$\mathcal{L}$$ be a relation called having a life which is the reflexive transitive symmetric closure of the relation on $$\mathcal{P}(\mathscr{X})\cup\{{\sf f}\}$$ associating all atoms corresponding to people without children to $${\sf f}$$. Now define $$\widehat{{\sf Par}}:\mathcal{P}(\mathscr{X})\to\mathcal{P}(\mathscr{X})\cup\{{\sf f}\}\big/\mathcal{Life}$$ $$\widehat{{\sf Chi}}:\mathcal{P}(\mathscr{X})\cup\{{\sf f}\}\big/\mathcal{Life}\to\mathcal{P}(\mathscr{X})$$ as before except have $$\widehat{\sf Par}$$ send the empty set to $$[{\sf f}]$$ and $$\widehat{\sf Chi}$$ send $$[{\sf f}]$$ to the empty set. Now, we genuinely have that $$\widehat{\sf Par}\dashv\widehat{\sf Chi}.$$ (All of this is well intended humor; my wife and I look forward to joining the ranks of parents with no free time.)

• Congrats and good luck! Those sleepless nights doubled my MO points. Jul 30, 2022 at 12:08
• @VilleSalo Thank you! Aug 2, 2022 at 0:52

If you're willing to admit your example as "describing a concept by an adjunction", then I would argue that any binary relation can be "described by an adjunction", namely the Galois connection that it induces.

To be sure, your example isn't quite the Galois connection induced by the relation of "is a parent of". For one thing, the latter is contravariant; it consists of

• $$\rm Par'$$ that sends a set $$C$$ of people to the set of people who are parents of everyone in $$C$$ (thus, $$\rm Par'(C)$$ is empty if $$C$$ doesn't consist of a group of siblings), and
• $$\rm Chi'$$ that sends a set $$P$$ of people to the set of people who are children of everyone in $$C$$ (thus, $$\rm Chi'(P)$$ is empty if $$P$$ has cardinality greater than 2, or is a pair of people who have never had children together).

However, suppose we consider instead the relation "is not a parent of". Then we get another contravariant Galois connection consisting of

• $$\rm Par''$$ that sends a set $$C$$ of people to the set of people $$x$$ such that no one in $$C$$ is a child of $$x$$.
• $$\rm Chi''$$ that sends a set $$P$$ of people to the set of people $$y$$ such that no one in $$P$$ is a parent of $$x$$.

Now compose this Galois connection on one side with the contravariant "complementation" automorphism of the set of sets of people, and we get a covariant adjunction consisting of

• $$\rm Par$$ that sends a set $$C$$ of people to the set of people who are a parent of at least one person in $$C$$.
• $$\rm Chi$$ that sends a set $$P$$ of people to the set of people both of whose parents are in $$P$$.

This is the same $$\rm Par$$ as yours, but a different $$\rm Chi$$. As appropriate for a covariant adjunction, we have $$C \subseteq {\rm Chi}({\rm Par}(C))$$ (if you are in the set $$C$$, then both of your parents are in the set of all parents of people in $$C$$) and $${\rm Par}({\rm Chi}(P)) \subseteq P$$ (any parent of someone both of whose parents are in $$P$$ must be in $$P$$). The childless are no exception: if no one in $$P$$ has any children, then $${\rm Chi}(P) = \emptyset$$, hence $${\rm Par}({\rm Chi}(P)) = \emptyset \subseteq P$$.

I could be misunderstanding, but I don't think your $$\rm Chi$$ is adjoint to $$\rm Par$$ in any sense, even after correcting for childlessness. You seem to be using the relations $$a\le g(f(a))$$ and $$b \le f(g(b))$$ that hold for a contravariant Galois connection but applying them to covariant functors, but I don't think that's any kind of adjunction: if you dualize one of the categories to make the functors covariant, you also need to flip the direction of one of the inequalities.

Anyway, since binary relations abound in the real world, so do the Galois connections they generate.

• Another way to see that the Par and Chi in the question aren't adjoints is that both preserve unions and neither preserves intersections. So both want to be the left adjoint and neither wants to be the right adjoint. That's not something covariant functors can do. Jul 31, 2022 at 16:36
• You are correct that I was incorrect about this being an adjunction, thank you. (my example was mostly in jest, but I did have a nagging feeling that I was wrong somewhere) Aug 1, 2022 at 1:41