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Praphulla Koushik

3:46 PM

Hello

symmetric_cow

Hi!

Praphulla Koushik

I am studying about stacks and gerbes

What’s your background?

I am the user with name cello

symmetric_cow

I'm interested in Gromov Witten theory so I'm trying to go through some notes on stacks

Praphulla Koushik

You have asked I read about stacks from Angelo Vistoli’s notes on Descent theory

what notes are you following

symmetric_cow

somewhat all over the place -- I wanted to get my hands on computations rather quickly so I was looking into Vistoli's intersection theory on stacks - recently I also found Kai Behrend has some notes on localisation and GW invariants (which starts by talking about stacks) so I might look into that a bit too

but I should look into vistoli's notes on descent theory too for some of the more technical aspects (descent) ...

Praphulla Koushik

3:56 PM

Yes.. you can start reading that and we can discuss if you have anything specific..

symmetric_cow

Sure

Praphulla Koushik

Btw what are stacks for you?

symmetric_cow

are you interested in stacks for the sake of it or for other reasons (e.g. maybe you're interested in enumerative geometry / or geometric rep theory etc. )?

uh a category fibred in groupoids where your isomorphism presheaf is a sheaf / all descent datum is effective?

or do you mean intuitively?

Praphulla Koushik

I am interested in concepts of gerbes.. they are stacks plus something so I was studying about stacks..

@symmetric_cow I also know have same understanding... I don’t know what can be an intuitive answer,,, do you have one?

symmetric_cow

4:18 PM

i see - i don't really know anything about gerbes

this might change once i read more -- but - at least - the intuition for a category fibred over groupoids is to think of vector bundles over topological spaces, so your CFG is the cat. of vector bundles F: C --> T = Top, and the axioms defining a CFG makes a lot of sense if you have X-->Y in top, and V in C where F(C) = Y, then there exists an object in C : W--> V , with F(W) = X - which is to say that you can pullback vector bundles

in moduli problems if you have a family of ___ over Y then you can pull this back to a family of ___ ov…

then there is something known as a deligne-mumford stack which shows up quite often in algebraic geometry -- which is a stack with some nice properties allowing you to talk about geometric properties of stacks , e.g. what does it mean for a morphism of (DM) stacks F-->G to be finite type etc.

so (like the joke goes) you can pretend this is a scheme except it's not a scheme..

Praphulla Koushik

this might change once i read more -- but - at least - the intuition for a category fibred over groupoids is to think of vector bundles over topological spaces, so your CFG is the cat. of vector bundles F: C --> T = Top, and the axioms defining a CFG makes a lot of sense if you have X-->Y in top, and V in C where F(C) = Y, then there exists an object in C : W--> V , with F(W) = X - which is to say that you can pullback vector bundles... yes, this is the first intuitive example I have seen..

I don’t understand what do you mean by “in moduli problems if you have a family of ___ over Y then you can pull this back to a family of ___ over X ”

symmetric_cow

haha sorry that might be a bit too vague --- are you familiar with moduli problems in algebraic geometry ?

Praphulla Koushik

No I wanted to study about that but did not get enoug motivation and necessity

There is a conference in Lisbon this September 26- October 6.. this is devoted to categories and stacks in algebraic geometry and algebraic topology.

symmetric_cow

i see
essentially what i said above - it's kind of the same idea as a vector bundle -- but instead of your fibres being vector spaces you can think of your fibres as being a geometric object, for example a genus 2 curve (riemann surface)

so the algebro/geometric analogue of a fibre bundle, and pullbacks etc. work the same way

unfortunately i will have started school by then !

Praphulla Koushik

@symmetric_cow grad school??

where!

symmetric_cow

4:29 PM

im in the states

and i think most schools in the US start in sept?

actually im not sure

but i do

Praphulla Koushik

Good luck :)