Timeline for Are Conway's combinatorial games the "monster model" of any familiar theory?
Current License: CC BY-SA 4.0
19 events
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| May 1, 2020 at 14:03 | comment | added | Philip Ehrlich | Tim@. I suspect you are right. | |
| May 1, 2020 at 14:02 | comment | added | Philip Ehrlich | Mike@. These are interesting questions to which I have not given any thought. | |
| May 1, 2020 at 6:09 | comment | added | მამუკა ჯიბლაძე | @TimCampion But in any case, while there are many partially ordered abelian groups that are injective for not-necessarily-embedding morphisms, the one injective for embeddings is unique up to isomorphism (and forcibly a proper class). And I believe it is also injective for not-necessarily-embeddings, no? | |
| May 1, 2020 at 3:59 | comment | added | Mike Battaglia | And, if they are also the universally embedding totally ordered free abelian group if you also count $\lt$... | |
| May 1, 2020 at 3:49 | comment | added | Mike Battaglia | @PhilipEhrlich, just saw in your edit that my question about the games being the universally embedding abelian group was answered in the affirmative in your edit. So I guess there's the same question for the surreals then, if they are the universally embedding free abelian group... | |
| May 1, 2020 at 3:38 | comment | added | Mike Battaglia | Thanks @PhilipEhrlich. So if the games are the "universally embedding" partially ordered abelian group, and given that every abelian group can be trivially made into a poset, does that mean it is also the universally embedding abelian group? Likewise, would the surreals be the universally embedding free abelian group? | |
| May 1, 2020 at 2:06 | vote | accept | Mike Battaglia | ||
| Apr 30, 2020 at 22:43 | comment | added | Philip Ehrlich | Tim Campion@. Thank you for the clarification. | |
| Apr 30, 2020 at 22:37 | comment | added | Tim Campion | @მამუკაჯიბლაძე Because being an embedding is a stronger condition than being a monomorphism, injectivity-with-respect-to-embeddings is also in a sense weaker than injectivity-with-respect-to-monomorphisms. Really we're just talking about injectivity in the category whose morphisms are embeddings. | |
| Apr 30, 2020 at 22:36 | comment | added | Tim Campion | @PhilipEhrlich I'm not questioning your answer. When I wrote $\mathbf{No}$ that was a brain-o for the class of all games. I'm just noting that the use of the term "monster model", which really goes back to the question statement, clashes with the standard usage in model theory. | |
| Apr 30, 2020 at 22:17 | comment | added | Philip Ehrlich | Tim Campion@. I still am not clear as to what point you are trying to make. Are you questioning my answer to the question. If so, on the basis of what? | |
| Apr 30, 2020 at 21:55 | comment | added | მამუკა ჯიბლაძე | @TimCampion This is stronger than injectivity (otherwise uniqueness would be problematic). This is because homomorphisms required in the statement satisfy $x\leqslant y$ iff $f(x)\leqslant f(y)$; in particular they are embeddings. | |
| Apr 30, 2020 at 20:05 | comment | added | Philip Ehrlich | Tim Campion@. I'm not sure I understand your point. The question was not about $\mathbf{No}$ but rather about the more general class of games. Moews's result like Lurie's is about the class of games. | |
| Apr 30, 2020 at 19:50 | comment | added | Tim Campion | If I understand the statement correctly, then I think model theorists would say that (the elementary theory of) $\mathbf{No}$ is the model completion of the theory of partially ordered abelian groups. But since the theory of partially ordered abelian groups is not complete, I don't think model theorists would call $\mathbf{No}$ "the monster model" of this theory -- an incomplete theory has one monster model for each completion. | |
| Apr 30, 2020 at 19:48 | comment | added | Tim Campion | Category theorists would probably say that a "universally embedding" partially ordered abelian group is a partially ordered abelian group which is injective with respect to embeddings. | |
| Apr 30, 2020 at 19:26 | history | edited | Philip Ehrlich | CC BY-SA 4.0 |
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| Apr 29, 2020 at 13:19 | history | edited | Philip Ehrlich | CC BY-SA 4.0 |
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| Apr 29, 2020 at 13:08 | comment | added | user44143 | I’m glad to learn that Lurie’s early work was concrete enough for me to understand and appreciate. | |
| Apr 29, 2020 at 12:53 | history | answered | Philip Ehrlich | CC BY-SA 4.0 |