Timeline for Is Joyal's category $\Theta_n$ the "only" Reedy category which is dense in $n$-categories?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2022 at 23:18 | vote | accept | Tim Campion | ||
| Sep 10, 2022 at 23:16 | answer | added | Tim Campion | timeline score: 4 | |
| Jan 16, 2022 at 16:00 | comment | added | Theo Johnson-Freyd | For some reason the construction "... something to say ... about getting one's nerves ..." continues to amuse me. | |
| Jan 16, 2022 at 15:23 | comment | added | Tim Campion | @SimonHenry Maybe it becomes more plausible if I insist on having an elegant Reedy category. In this case, the Reedy factorization system must have the left half be the split epis, and the right half is almost forced to be the monos. I don't see how to cook up dense categories of polygraphs among whose free $n$-categories every surjection splits... | |
| Jan 16, 2022 at 15:18 | comment | added | Simon Henry | I would be very surprised if something like this was true. The category $\Theta$ is "Canonical" only if you start from the globular point of view on higher categories. But you can use other combinatorics to describe Higher categories (simplicial, cubical, or more complicated things using polygraphs) which own generalized Nerves style description of higher categories. Of course it is a bit tricky to see in advance if these are Reedy categories or not (it is not so obvious that Theta is Reedy for example) but I really don't expect that none of these would yield Reedy categories... | |
| Jan 16, 2022 at 15:06 | comment | added | Tim Campion | @ZhenLin Oh wow, of course! Probably the subcategory of $\Theta_n$ where you build up wreath products with $\Delta_{\leq [2]}$ rather than $\Delta$ is already dense in strict $n$-categories too! This points to the question being subtler than I anticipated, and hinging crucially on the weak / strict distinction. In light of this, I'm not quite sure how I should modify the question, but tentatively I should probably, as you suggest, just restrict attention to the weak case. | |
| Jan 16, 2022 at 15:02 | comment | added | Zhen Lin | Isn't $\mathbf{\Delta}_{\le 2}$ dense in $\textbf{Cat}$? I think you need to restrict attention to $(\infty, n)$-categories to justify the "need" for higher simplices. | |
| Jan 16, 2022 at 14:43 | history | asked | Tim Campion | CC BY-SA 4.0 |