So okay, here's a thread on the category of finite sets and a way in which it controls algebraic structure in symmetric monoidal categories. I think it's some really pretty stuff.

First let's give some terminology and definitions. I'm going to let Γ denote the category whose objects are finite sets and morphisms are functions of finite sets. There are a whole lot of finite sets so to simplify thinking about this category, I'll work with a "skeleton."

What that means is that I'll just work with isomorphism classes of objects of Γ, i.e. if there is a bijection between two finite sets A and B, I'll think of them as the same set (this can be made categorically precise).

What I end up with then is a category whose objects are countable, namely the set {∅, {1}, {1,2}, {1,2,3},...} and whose morphisms are just functions between these sets. This will be enough to get at some cool algebraic structure.

At some point I might start writing 1 for {1} and 2 for {1,2}, and n for {1,2,3,...,n} but I'll try to let you know that I'm doing it, and hopefully it won't be too confusing of a notational switch.