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Background for the curious reader:

An ordinal $\beta$ is a transitive set in the sense that $\alpha\in\beta$ implies $\alpha\subset\beta$. Any ordinal is naturally well-ordered under $\in$ (so any subset of it has a least element), and any well-order is isomorphic to an ordinal. In fact, any class (naively, collection) of ordinals is itself well-ordered under the relation $\in$. This fact allows for the usage of transfinite induction and transfinite recursion.

We have three types of ordinals: the empty set $0=\emptyset=\{\; \}$, successor ordinals $S(\alpha)=\alpha\cup\{\alpha\}$ where $\alpha$ is an ordinal, and limit ordinals, which are all the other ones. Finite ordinals are either $0$ or successors, the set $\omega=\{0,1,2,\dots\}$ is a limit ordinal. A limit ordinal $\alpha$ has the property that $\alpha=\sup_{\beta<\alpha}\{\beta\}=\bigcup_{\beta<\alpha}\beta$.

My question

Ordinal arithmetic can be defined recursively as follows:

  1. $\alpha+0=\alpha$, $\alpha+S(\beta)=S(\alpha+\beta)$, $\alpha+\sup_{\gamma<\beta}\{\gamma\}=\sup_{\gamma<\beta}\{\alpha+\gamma\}$;
  2. $\alpha\cdot0=\alpha$, $\alpha\cdot S(\beta)=\alpha\cdot\beta+\alpha$, $\alpha\cdot\sup_{\gamma<\beta}\{\gamma\}=\sup_{\gamma<\beta}\{\alpha\cdot\gamma\}$;
  3. $\alpha^0=1$, $\alpha^{S(\beta)}=\alpha^\beta\cdot\alpha$, $\alpha^{\sup_{\gamma<\beta}\{\gamma\}}=\sup_{\gamma<\beta}\{\alpha^\gamma\}$.

Alternatively, one can define:

  1. $\alpha+\beta$ is the unique ordinal isomorphic to the disjoint union $\{0\}\times\alpha\cup\{1\}\times\beta$ given the lexicographic order.
  2. $\alpha\cdot\beta$ is the unique ordinal isomorphic to the Cartesian product $\beta\times\alpha$ given the lexicographic order.

As the disjoint union and Cartesian product are simply the categorical coproduct and the categorical product, I wonder if there is some way to actually categorify these alternate definitions. Additionally, I am not aware of any non-recursive version of exponentiation, so I would be curious if a categorical formulation of addition and product of ordinals also allows for a categorical (hence non-recursive) formulation of exponentiation.

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    What do you mean exactly by categorifying these definitions? What are you looking for?2013-05-05
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    I want the sum, product, and exponential to be objects in some category that includes all the ordinals, and which satisfy some universal properties in some category. The question is which category and which properties?2013-05-08
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    Vladimir, just a thought: if we view ordinals as being elements of the skeleton category of totally ordered sets, perhaps coproducts of ordinals coincide with the usual ordinal sum. I could be wrong, though. Good question, by the way.2013-07-01

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