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Stream: learning: questions

Topic: groupoid representations and Erlangen geometry

Matteo Capucci (he/him) (Feb 17 2022 at 12:16):

Is anyone aware of a relationship between

representations of a given groupoid $\mathcal S$ of symmetries, like the general $k$ -linear groupoid $\Sigma_{n \in \N} \mathbb BGL_k(n)$
the related 'Erlangen' geometry, that is the category of spaces with $\mathcal S$ -symmetries, like the category of $k$ -vector spaces

The category (1) is the category of spaces invariant (or equivariant) under $\mathcal S$ transformations, but you can't map between spaces with symmetries from different connected components of $\mathcal S$ . So it doesn't fully recover (2), but there's should be some sense it which the two are related, or some other way to relate a given groupoid of symmetries with spaces invariant under those symmetries.

Reid Barton (Feb 17 2022 at 12:35):

It looks like (1) is [a skeleton of] the subcategory of isomorphisms in (2) [assuming we restrict (2) to finite-dimensional vector spaces]. Is that what you're looking for?

Matteo Capucci (he/him) (Feb 17 2022 at 13:02):

Mmh indeed, good point

Matteo Capucci (he/him) (Feb 17 2022 at 13:02):

But unfortunately it's not what I'm looking for. I'd like to recover the entire category!

John Baez (Feb 17 2022 at 15:51):

Traditionally, or even 'neo-traditionally', Klein geometry considers groups and groupoids, not categories.

John Baez (Feb 17 2022 at 15:54):

For example, James Dolan has a lot of ideas about Klein geometry and groupoids. I described these in week 249 and some following issues of This Week's Finds.

John Baez (Feb 17 2022 at 15:55):

At the time he said "I'm not really a category theorist - I'm a groupoid theorist".

Matteo Capucci (he/him) (Feb 17 2022 at 16:39):

John Baez said:

Traditionally, or even 'neo-traditionally', Klein geometry considers groups and groupoids, not categories.

+1 for 'neo-traditionally'

Matteo Capucci (he/him) (Feb 17 2022 at 17:53):

(also your link seems relevant, but I still have to read it -- will come back soon)

John Baez (Feb 17 2022 at 18:05):

Yeah, I think anyone interested in groupoids and Klein's Erlangen program should learn this stuff that James Dolan did (in conversations with me and Todd Trimble).

Matteo Capucci (he/him) (Feb 17 2022 at 22:31):

So I was intrigued by this:
image.png

Matteo Capucci (he/him) (Feb 17 2022 at 22:32):

but then I don't really see this development in the subsequent weeks! I've got a bit lost

Matteo Capucci (he/him) (Feb 17 2022 at 22:34):

I see you get to define Hecke operators from spans of groupoids in week 254, could it be that 'give' above means 'you can get a'? So not all vector spaces and not all linear operators arise like that?

Matteo Capucci (he/him) (Feb 17 2022 at 22:38):

Mmh maybe this is it image.png

Matteo Capucci (he/him) (Feb 17 2022 at 22:39):

so I'm back to the drawing board!

John Baez (Feb 17 2022 at 23:36):

Yes, this is the construction: a groupoid X gives a vector space [X] whose basis consists of isomorphism classes of objects in X, and a span from X to Y gives a linear operator from [X] to [Y]. I explain this in week256:

So, let's finally figure out how a span of groupoids
                   S
                  / \
                 /   \
                /     \
               v       v
              X         Y
gives a linear operator

[S]: [X] → [Y]

John Baez (Feb 17 2022 at 23:39):

But I should have pointed you to this:

Groupoidification

which has the whole Tale of Groupoidification in one place, and also two papers that go into more detail: "Groupoidification made easy" and "Higher-dimensional algebra VII: groupoidification."