Timeline for "the" random permutation
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 19, 2019 at 23:00 | comment | added | YCor | "*«permutation» here is an even weirder term": 1 century ago "permutation" rather had this meaning of a pair of total orderings, and the corresponding function (now called permutation) was rather called "substitution". So I think it now sounds weird, but is not weird, just a trace of ancient terminology. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 6, 2014 at 16:44 | comment | added | Noah Schweber | In general, I think that things are only guaranteed to work at the countable level (and maybe at the level of $\omega_1$, too). | |
| Jun 6, 2014 at 16:41 | comment | added | Noah Schweber | "For every $\kappa$, there is a $\Sigma$-structure $S$ of cardinality $\kappa$ with age $\{$finite permutations$\}$ which is 'very homogeneous'." Then we could say that we had a functor from the category of infinite sets (or, more compactly, the category of infinite cardinals) to the category of $\Sigma$-structures up to isomorphism that was the "random permutation builder." However, in order for this to actually be functorial, we need that in fact the structure associated to $\kappa$ be $<\kappa$-homogeneous, and it's not at all clear to me that we can necessarily do this in ZFC. (Can we?) | |
| Jun 6, 2014 at 16:39 | comment | added | Noah Schweber | How could there be? There isn't even a canonical way to produce a random permutation with domain $\mathbb{N}$: the construction of the random permutation requires us to make (infinitely many) choices along the way. Maybe the target category could be "structures (in the signature $\Sigma$ of the random permutation) up to isomorphism," in which case the morphisms would presumably be "homomorphisms up to isomorphism." I suspect this would get a bit weird, but moving on: we would then want the following claim to be true: (cont'd) | |
| Jun 6, 2014 at 12:44 | comment | added | john mangual | My question is on whether random permutations can be thought of categorically. I was hoping maybe there was some functor Set to some enriched version of Set with a uniform random permutation associated to each set. | |
| Jun 6, 2014 at 0:57 | comment | added | Noah Schweber | I just noticed that Qiaochu Yuan already linked to the $n$-category cafe post; oh well. | |
| Jun 6, 2014 at 0:56 | history | answered | Noah Schweber | CC BY-SA 3.0 |