(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\\).