Ordinals in category theory?

Question

I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $\geq \omega$.

Through the basic notions of category theory, what is an ordinal? And what does ordinal arithmetic look like categorically?

Answer 1

The class $\bf Ord$ of ordinal numbers can be described as the free algebra on a singleton, with respect to a certain monad $P$ on the category of classes (although not precisely a $P$-algebra).

The reference is this old paper by Rosicky.

Answer 2

You may be interested in the concept of a natural numbers object in a category. I find this to be a beautifully simple idea - it basically implements the concept of zero and a successor function. If you're working in a category like Set where such an object exists, you can think about things like iterating a function $n$ times.