# A categorical characterization of ordinal numbers

It's rather easy to notice that the operation of join of categories reproduces the ordinal sum once restricted to act on (iso classes of) well-ordered set; it's rather easy to see that $\alpha\star [0]$ (as a category) equals $\alpha+1$ (as number).

And in fact there is more ([Jec, p. 16]): the transfinite sequence $s\colon \mathbf{Ord}_{<\alpha}\to X\colon \langle a_\xi\mid \xi < \alpha\rangle$, extended with value $x_0$ on the successor $\alpha+1=\alpha^+$, corresponds exactly to the operation which, given a category $\cal C$, adds a (strained) terminal object $*$ defining $\mathcal C^\rhd = \mathcal C\star [0]$($=\mathcal C\coprod\{*\}$ with a unique arrow between any $C\in\cal C$ and $*$), and given a functor $F\colon \mathcal C\to \cal X$, it is extended with value $X_0\in\cal X$ on the unique object $*$ of $[0]$, coinciding with $F$ on $\cal C$.

Now: $\bf Cat$ endowed with the bifunctor $\star$ becomes monoidal [Joy, pp. 26-27] and the structure is closed on both sides (albeit not biclosed; the $\star$ is not symmetric)

Should this give a characterization of ordinals as some sort of cocompletion of the free $\star$-monoidal category on one generator? Can something non trivial be desumed by this fact?

Edit: in fact as it is stated the question is unclear. You must first complete, and then close under $\star[0]$ (I was confused, and maybe I still am, by the fact that $\star[0]$ commutes with colimits).

[Jec] Jech, Thomas. Set theory. Vol. 79. New York: Academic press, 1978.

• The operation $\star$ does not commute with colimit: indeed if that where true then for every limit ordinal $\lambda=\bigcup_{\gamma < \lambda} \gamma$ you should have that $\lambda+1=\lambda\star[0]=\bigcup_{\gamma < \lambda} \gamma\star[0]=\bigcup_{\gamma < \lambda}\gamma+1=\lambda$ things that's not possible since $\lambda$ doesn't have a max while $\lambda+1$ does. Commented May 20, 2014 at 14:34
• Note that in the note you have linked Joyal define $\star B$ as a functor of type $\mathbf {Cat} \to B\setminus \mathbf {Cat}$ not $\mathbf {Cat} \to \mathbf {Cat}$. Commented May 20, 2014 at 14:36
• I guess that for help could be useful to characterize $\star$ by universal property, otherwise is not clear what a category (not necessarily a subcategory of $\mathbf{Cat}$) closed by $\star$ should be. Commented May 21, 2014 at 11:10

## 1 Answer

You're not going to learn much about the conceptual or categorical structure of the ordinals from their classical presentation, not just because Excluded Middle is needed at every stage but because of normalformitis: the systematic elimination of structure.

That being said, there is a paper by Peter Johnstone, A topos-theorist looks at dilators (JPAA, 58, 1989) that applied Yves Diers' extensive work on categories with multiple-valued adjoints to Jean-Yves Girard's $\Pi^1_2$ Logic: Part I, Dilators (Ann. Math. Logic 21, 1981). This studied ordinal-building constructions considered as functors that preserve pullbacks.

You cite Andre Joyal. He wrote a book with Ieke Moerdijk called Algebraic Set Theory (CUP 1995) and the subject was taken up by others and presented on Steve Awodey's website.

They characterised the universes of sets ($\in$-strutures in this context) and ordinals as free algebras for arbitrary unions and a unary operation $s$ (singleton or successor) subject to various equations:

• for sets, no condition,
• for thin ordinals, $x\leq s x$,
• for plump ordinals, $x\leq y\Longrightarrow s x\leq s y$,
• for directed ordinals, $s x\lor s y= s(x\lor y)$,

using my terminology for the ordinals.

Earlier in the history of elementary toposes, Gerhard Osius wrote about Categorical Set Theory: a characterisation of the Category of Sets (JPAA, 4, 1974) and I picked up his ideas in my Intuitionistic Sets and Ordinals (JSL, 61, 1996). My work was later taken up by Varmo Vene and others with application to process algebra.

In this setting, the membership relation is written $\in:X\to P X$ and so considered as a coalgebra for the powerset or other functors. The important properties are extensionality and well-foundedness. The former is interpreted categorically by saying that the map $\in$ is mono and the second by a "broken pullback" diagram that you can find in my paper or my book.

The subset relation and the notion of simulation in process algebra are coalgebra homomorphisms. Recursion over ordinals was expressed by various people, including Adam Eppendahl, as coalgebra-to-algebra homomorphisms.

By applying these ideas in different categories, such as posets and binary semilattices, and with different notions of monomorphism, we obtain the four kinds of sets and ordinals above and others. These applications are summarised in Sections 6.3 and 6.7 of my book Practical Foundations of Mathematics (CUP 1999).

• I know about Joyal-Moerdijk book. I'm also caught by this: "You're not going to learn much about the conceptual or categorical structure of the ordinals [...] because of normalformitis: the systematic elimination of structure." Please, expand! Commented May 22, 2014 at 13:08
• Linking my answer more closely to your quesion, your join of categories is an example of a dilator, whilst the Joyal-Moerdijk approach provides the universal property for which you are looking. By normalformitis I mean that very strong axioms such as classical logic "tidy away" most of the"complications" in a structure, leaving something that is uninformative if you want to understand what's going on. Higher ordinal notations, for which Girard introduced dilators, so show you a bit more. Commented May 22, 2014 at 15:08
• However, if I ever return to this topic it will be to look for extensional well founded coalgebras in different categories. Fixed point objects should be ordinals in domains. The Conway numbers are some kind of two-sided ordinals and the Dedekind reals should be their domain-theoretic analogue. Commented May 22, 2014 at 15:10
• I found this archive.numdam.org/ARCHIVE/DIA/DIA_1979__2_/DIA_1979__2__A5_0/… old paper by Rosicky; how does the fact that $Ord$ can be regarded as the free "nonempty subsets"-algebra on the signleton fit into your answer? Is this result present (or maybe generalized) in Joyal-Moerdijk? Commented May 28, 2014 at 20:23
• There is now a draft paper called "Ordinals as Coalgebras", with new work, on paultaylor.eu/ordinals Commented Sep 6, 2023 at 13:24