Timeline for What are some examples of weak ω-categories?
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| May 8, 2013 at 4:30 | comment | added | Ricardo Andrade | @Jeremy: Thank you for the clarification. | |
| May 2, 2013 at 23:41 | comment | added | Jeremy Hahn | Yes it seems you have the idea. The higher identity morphisms should indeed be degenerate multi-simplices or globules. Even in the case of complicial sets the higher identity morphisms should be characterized by degeneracy conditions rather than thinness. | |
| May 2, 2013 at 19:14 | comment | added | Ricardo Andrade | @Jeremy: Thank you very much for the explanation. I just want to make sure I understand correctly what you mean by "higher identity morphism". Would that be a degenerate simplex (or globule, etc) in some multi-simplicial or globular approach? Or maybe it is a thin simplex in a complicial approach? Is that correct? Is there a better way to think about it? | |
| May 2, 2013 at 16:57 | comment | added | Jeremy Hahn | @Ricardo I can try... it's hard without pictures. Basically, in the first homotopy theory a morphism is invertible if it is part of a tower of adjoints that terminates at a higher identity morphism. This is in contrast to the second theory where a morphism is invertible simply if it is part of a tower of adjoints, with no restriction that the tower eventually degenerate into higher identity morphisms. | |
| Apr 30, 2013 at 18:36 | comment | added | Ricardo Andrade | @Jeremy Hahn: Can you describe what an invertible morphism in a (weak) $\omega$-category would look like in the first homotopy theory you consider? | |
| Apr 30, 2013 at 15:52 | comment | added | David Ben-Zvi | An example I think of Dmitri Pavlov's comment: spaces are $E_\infty$ algebras in the category of spaces with correspondences, and the corresponding Morita $(\infty,\infty)$ category should have objects spaces, morphisms correspondences, 2-morphisms correspondences of correspondences, and so on all the way up. OTOH I think any space (or $E_\infty$ algebra) is $n$-dualizable for any $n$, so not sure how this agrees with Sam's comment. | |
| Apr 30, 2013 at 13:33 | history | edited | Omar Antolín-Camarena | CC BY-SA 3.0 |
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| Apr 28, 2013 at 6:17 | comment | added | S. Carnahan♦ | I agree that your question looks like a duplicate, but I am hesitant to close it unilaterally. | |
| Apr 28, 2013 at 1:55 | history | edited | Omar Antolín-Camarena | CC BY-SA 3.0 |
added previous question of which this is a duplicate; deleted 258 characters in body
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| Apr 27, 2013 at 15:50 | comment | added | Jeremy Hahn | I think the cobordism categories are much more interesting in the first homotopy theory, the advantage of the second being that it has a nice "coinductive" definition ala the model structure on strict omega categories. Presumably strict omega categories have a second model structure that better represents the first theory. | |
| Apr 27, 2013 at 15:46 | comment | added | Jeremy Hahn | I meant infinity groupoid. Homotopy inverses should always count as adjoints, but only in the second theory is there a kind of converse for towers of adjoints going all the way up to infinity. | |
| Apr 27, 2013 at 14:40 | comment | added | Dmitri Pavlov | This might be too naïve, but E_n-algebras can be organized into an (∞,n+1)-category, so perhaps E_∞-algebras can be organized into an (∞,∞)-category? | |
| Apr 27, 2013 at 12:51 | comment | added | Omar Antolín-Camarena | Did you mean $\infty$-groupoid instead of contractible, @JeremyHahn? I mean an ordinary groupoid has all adjoints, right? I certainly don't want it to be contractible as an $(\infty,\infty)$-category. (Or do I?) If this second homotopy theory of $(\infty,\infty)$-categories makes either (1) a all homotopy types contractible, or (2) homotopy inverses not count as adjoints, it doesn't sound like such a great idea. | |
| Apr 27, 2013 at 4:46 | comment | added | Jeremy Hahn | Omar it is not clear what the homotopy theory of (infinity,infinity)-categories should be. Let J denote the nerve of the contractible groupoid with 2 objects. Then inverting J and all of its suspensions yields one homotopy theory. There is a second, different homotopy theory in which a category with all adjoints is contractible. If one looks only at (infinity,n)-categories these two homotopy theories are the same, but I believe they are not the same at (infinity,infinity). I think that the category of bordisms is only contractible in the second sense | |
| Apr 27, 2013 at 2:30 | comment | added | Omar Antolín-Camarena | Yes, I think that's right @SamGunningham. I've added a remark about this to the question. | |
| Apr 27, 2013 at 2:29 | history | edited | Omar Antolín-Camarena | CC BY-SA 3.0 |
incorporated Sam Gunningham's correction
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| Apr 26, 2013 at 22:49 | history | edited | Ricardo Andrade |
added tag; edited tags
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| Apr 26, 2013 at 21:13 | comment | added | Sam Gunningham | I thought the $(\infty, \infty)$-category of bordisms should in fact be an $(\infty, 0)$-category. Naively, wouldn't being ``$\infty$-dualizable'' mean that every morphism is invertible? | |
| Apr 26, 2013 at 20:10 | history | asked | Omar Antolín-Camarena | CC BY-SA 3.0 |