Timeline for Is there an accepted definition of $(\infty,\infty)$ category?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2023 at 18:32 | comment | added | Theo Johnson-Freyd | I wandered back to this after 10 years. Yes, I convinced myself that $\mathbf{LCat}_{(\infty,\infty)} \to \mathbf{Cat}_{(\infty,\infty)}$ is a coreflective inclusion. I think it's formal and has nothing to do with the specifics: I think it uses only that $\mathbf{Cat}_{(\infty,n)} \to \mathbf{Cat}_{(\infty,n+1)}$ is fully faithful. The question is then whether the colocalization $\mathbf{Cat}){(\infty,\infty)} \to \mathbf{LCat}_{(\infty,\infty)}$ localizes only against this "Cheng axiom" (walking $\infty$-adjunction) $\to$ (pt), or if there are other relations. | |
| Jun 14, 2021 at 23:01 | comment | added | Andrea Marino | Is there some similar phenomena for $n \to \infty$ in case one wants to model (the homotopy theory of) $(\infty, 1) $-categories in terms of $n$-categories? If yes, in this case one uses the left tower or the right tower to send $n\to \infty$? Sorry but I am not really familiar with $n$-categories if $1 < n < \infty$... | |
| Dec 15, 2017 at 16:49 | history | edited | John Baez | CC BY-SA 3.0 |
is a sequences of |-> is a sequence of
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| Jun 22, 2013 at 21:57 | vote | accept | Theo Johnson-Freyd | ||
| Jun 21, 2013 at 19:28 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
added 1 characters in body
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| Jun 21, 2013 at 15:39 | comment | added | Chris Schommer-Pries | Hi Theo, I was traveling, so haven't had time for MO the past couple days. I edited my question a bit to make it more clear how I was try to address your question. | |
| Jun 21, 2013 at 15:36 | history | edited | Chris Schommer-Pries | CC BY-SA 3.0 |
Added a bunch more content to the answer.
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| Jun 20, 2013 at 19:39 | comment | added | Clark Barwick | I won't speak for Chris, but his and Charles's answers together seem to answer your question: One can construct homotopy theories that deserve the mantle $\mathbf{Cat}_{\infty}$ by taking a limit of "approximations" by homotopy theories of $(\infty,n)$-categories, as you suggest. The example of cobordism categories shows that there are at least two distinct homotopy theories that can be built this way. If you accept one of these two options as "official," the Unicity Theorem implies it is unique up to equivalence, regardless of which models of $(\infty,n)$-categories you approximate with. | |
| Jun 19, 2013 at 19:20 | comment | added | Theo Johnson-Freyd | So I take it your answer to my question is "No, there is not an accepted definition of $(\infty,\infty)$-category: there are two of them." Did I interpret you correctly? | |
| Jun 19, 2013 at 4:57 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
corrected several spelling errors
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| Jun 19, 2013 at 0:14 | history | edited | Chris Schommer-Pries | CC BY-SA 3.0 |
more comments at the end.
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| Jun 18, 2013 at 23:42 | history | answered | Chris Schommer-Pries | CC BY-SA 3.0 |