(I use this symbol to let beginners know that this is more difficult material, and that they shouldn't feel bad if it makes no sense to them!)

Ken wrote:

> The construction of \$$P \vee Q\$$ reminds me very much of how free categories, monoids, monads etc are constructed (i.e. induction), might it be regarded as some kind of free coproduct or something?

Anindya wrote:

> I'm guessing there's some deep categorical reason for this, i.e. at some level the two constructions are basically the same.

That's a great point! A partition is the same as an equivalence relation, and an equivalence relation on a set \$$X\$$ is the same as a groupoid that's also a preorder with \$$X\$$ as its set of objects. So, I believe when we're forming \$$P \vee Q\$$ we're forming the coproduct in the category of "groupoids that are also preorders with \$$X\$$ as set of objects". These things are pretty different than monoids, but there's a similarity: both are categories with a fixed set of objects. (A monoid is a category with one object.) I believe this makes them similar enough that the coproduct (aka \$$\vee\$$) of partitions is defined using the same sort of "alternating, inductive" construction as coproduct of monoids!

It's nice to look at coproducts in the category \$$\mathrm{Cat}/X\$$ where objects are "categories with \$$X\$$ as their set of objects" and morphisms are "functors that are the identity on objects". This gives that alternating inductive construction in a very pure form. Given categories \$$C\$$ and \$$D\$$ in \$$\mathrm{Cat}/X\$$, their coproduct is a category whose morphisms are alternating strings of morphisms in \$$C\$$ and \$$D\$$.