A groupoid has objects (points) and morphisms (arrows between points). We can "compose" morphisms f: x→y and g: y→z and get a morphism gf:x→z. Composition is associative. Each object x has an identity for composition, 1_x: x→x. Morphisms have inverses. That's all! (2/n)
-
-
Prikaži ovu nit
-
A set is the same as a groupoid with only identity morphisms. So, it has objects (points) but no interesting morphisms. A group is the same as a groupoid with just one object x. The elements of our group are the morphisms f:x→x. So you already know some groupoids. (3/n)
Prikaži ovu nit -
Around 2000 I figured out how to define the cardinality of a groupoid. For a groupoid that's just a set, the cardinality is just the number of objects. But throwing in extra morphisms *reduces* the cardinality! (4/n)
Prikaži ovu nit -
If you have a group G, you get a groupoid with one object: call it BG. Elements of G give morphisms in BG. The cardinality of BG is one over the usual cardinality of G: |BG| = 1/|G| So, the more morphisms BG has, the smaller its cardinality gets! (5/n)
Prikaži ovu nit -
Why should this be? It's because morphisms in a groupoid are 'ways for objects to be the same' (or 'isomorphic'). If an object is the same as itself in a lot of ways, it acts smaller, as it it were 'folded over'. That's the vague intuition, anyway. (6/n)
Prikaži ovu nit -
It turns out that nature knows about this! In particle physics, we compute the amplitude for particles to interact by summing over Feynman diagrams. But if a diagram has symmetries, we have to *divide by the number of symmetries* to get the right answer! (7/n)pic.twitter.com/gqvmTcktld
Prikaži ovu nit -
So there's not really a set of Feynman diagrams - there's a groupoid of them. And groupoid cardinality is not just crazy made-up shit. It's part of how nature works!!! More: https://arxiv.org/abs/math/0004133 … I've just learned something else about groupoid cardinality. (8/n)
Prikaži ovu nit -
I've been studying random structures in combinatorics. Like: if you have a random permutation of a 7-element set, how many cycles does it have, on average? These averages are often fractions. And it turns out they're often groupoid cardinalities! (9/n)pic.twitter.com/xUzsFoNNqS
Prikaži ovu nit -
It should have been obvious, since when you average over permutations you divide by n!, which is the number of elements of a permutation group. But now I see that many formulas for averages in combinatorics are secretly equivalences between groupoids! (10/n)
Prikaži ovu nit -
In combinatorics there's an idea called "bijective proof". To prove an equation between natural numbers, like n+1 choose k = n choose k + n choose k-1 you set up a 1-1 correspondence between finite sets. But what about equations between fractions? (11/n)
Prikaži ovu nit -
You can sometimes prove equations between fractions by setting up an equivalence between groupoids! Equivalent groupoids have the same cardinality. So I've been doing this: https://golem.ph.utexas.edu/category/2019/12/random_permutations_part_10.html … and it's lots of fun. (12/n)
Prikaži ovu nit -
For more, try Qiaochu Yuan's article on groupoid cardinality: https://qchu.wordpress.com/2012/11/08/groupoid-cardinality/ … and my article connecting it to the Riemann zeta function: http://math.ucr.edu/home/baez/week300.html … I still like the paper where groupoid cardinality was invented: https://arxiv.org/abs/math/0004133 … (13/n, n=13)pic.twitter.com/qN0PPki1QE
Prikaži ovu nit
Kraj razgovora
Novi razgovor -
Čini se da učitavanje traje već neko vrijeme.
Twitter je možda preopterećen ili ima kratkotrajnih poteškoća u radu. Pokušajte ponovno ili potražite dodatne informacije u odjeljku Status Twittera.