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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?

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  • $\begingroup$ Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup. $\endgroup$ – Qiaochu Yuan Jul 24 '18 at 4:26
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    $\begingroup$ Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories $\endgroup$ – Qiaochu Yuan Jul 24 '18 at 6:07
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    $\begingroup$ Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure? $\endgroup$ – Pece Jul 24 '18 at 6:43
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    $\begingroup$ @Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $\omega$ will have to be defined up to isomorphism of posets. I see no issue with that. $\endgroup$ – Kevin Carlson Jul 24 '18 at 17:00
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    $\begingroup$ @KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $\omega$ then. $\endgroup$ – Pece Jul 25 '18 at 8:31
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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.

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