Timeline for A categorical characterization of ordinal numbers
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 6, 2023 at 13:24 | comment | added | Paul Taylor | There is now a draft paper called "Ordinals as Coalgebras", with new work, on paultaylor.eu/ordinals | |
| May 29, 2014 at 14:23 | comment | added | Paul Taylor | The idea in the Rosicky paper (algebras for the "non-empty powerset" functor) looks a little like the application of my coalgebra ideas to directed ordinals. However, classical logic is too blunt an instrument to investigate this. Have you talked about this with any of the logicians in Padova? If you want to discuss it further with me, please do so privately by email. | |
| May 28, 2014 at 20:23 | comment | added | fosco | 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? | |
| May 22, 2014 at 17:39 | vote | accept | fosco | ||
| May 22, 2014 at 15:10 | comment | added | Paul Taylor | 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. | |
| May 22, 2014 at 15:08 | comment | added | Paul Taylor | 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. | |
| May 22, 2014 at 13:08 | comment | added | fosco | 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! | |
| May 21, 2014 at 8:18 | history | answered | Paul Taylor | CC BY-SA 3.0 |