Timeline for Removing the symmetry maps from a small category of cubes
Current License: CC BY-SA 4.0
16 events
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| May 20, 2023 at 1:08 | vote | accept | Philippe Gaucher | ||
| May 19, 2023 at 15:30 | answer | added | Philippe Gaucher | timeline score: 0 | |
| May 19, 2023 at 14:26 | comment | added | darij grinberg | One thing you can do is work with the sorting maps. The $i$-th sorting map $r_i$ changes the entries $x_i$ to $x_{i+1}$ to $\min\left\{x_i,x_{i+1}\right\}$ and $\max\left\{x_i,x_{i+1}\right\}$, respectively. The sorting maps generate a copy of the 0-Hecke monoid, if my memory doesn't cheat me. You can look at the category that they form when combined with the cofaces. This might be a smaller category than the one you're looking for, but well worth some study. | |
| May 19, 2023 at 13:56 | comment | added | darij grinberg | Ah, those are the $[n]$s here! Interesting, but yes, the notation should be better... | |
| May 19, 2023 at 8:37 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| May 19, 2023 at 8:32 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| May 19, 2023 at 8:26 | comment | added | Philippe Gaucher | @PeterLeFanuLumsdaine Probably $[1]^n$ is better than $[n]$ indeed. I use a different notations for presheaves. | |
| May 19, 2023 at 7:50 | comment | added | Peter LeFanu Lumsdaine | This is a nice question, but may I suggest two notational changes might make it clearer? Denoting the cube posets $[1]^n$ or $I^n$ rather than $[n]$ (both certainly occur in the cubical sets literature, but elsewhere $[n]$ is much more widely used for $\{0,…,n\}$, which the cubical sense clashes badly with); and using something like $\square'$ instead of $\widehat{\square}$, since $\widehat{-}$ is standard notation for presheaf categories, and in particular $\widehat{\square}$ is commonly used for the category of cubical sets, so it’s very confusing to see it meaning a small cube category? | |
| May 19, 2023 at 6:21 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| May 19, 2023 at 6:20 | comment | added | Philippe Gaucher | @darijgrinberg $[n]=\{0<1\}^n$, not $\{1<\dots<n\}$. For example, the map $(x_1,x_2)\mapsto (x_2,x_1)$ is strictly increasing. Poset means partially ordered set. | |
| May 19, 2023 at 1:59 | comment | added | darij grinberg | By "posets" you mean finite sets? And how do you get symmetries in that category? And isn't the only strictly increasing map $f : [n] \to [n]$ the identity? | |
| May 19, 2023 at 1:58 | history | edited | darij grinberg | CC BY-SA 4.0 |
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| May 18, 2023 at 3:03 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| May 18, 2023 at 2:55 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| May 16, 2023 at 15:06 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| May 16, 2023 at 14:59 | history | asked | Philippe Gaucher | CC BY-SA 4.0 |