Metrics on categories

Stream: maths

[Matthew Ballard]

We had a talk by Neeman yesterday. Formalizing the completion of a category with respect to a metric seems like a reasonable target. Heck triangulated categories are in reasonably in scope now. For background, see Neeman’s notes

[Matthew Ballard]

We’ve said it before: in this free world of ours there is no law prohibiting a mathematician from making up a long sequence of absurd-looking definitions.

[Adam Topaz]

It sounds cool (and actually kind of related to some discussions we had in LTE) but are there any applications you have in mind?

[Matthew Ballard]

GAGA?

[Adam Topaz]

how does GAGA fit into the picture?

[Matthew Ballard]

https://arxiv.org/abs/1804.01976

[Matthew Ballard]

Super cute

[Matthew Ballard]

Of course it would blow up the scope by orders of magnitude to just get these as categories

[Kevin Buzzard]

Note that my PhD student @Jujian Zhang is actively thinking about how to approach GAGA right now.

[Adam Topaz]

Locally-ringed G-spaces is G-spaces in the sense of rigid geometry?

[Matthew Ballard]

I think those are subset but confess to knowing nothing more than what is in the paper

[Adam Topaz]

Also I don't really see where metrics on categories comes into play in that GAGA paper...

[Matthew Ballard]

Metrics let you prove that certain functors out of $D^b(\operatorname{coh} X)$ are representable. Now you apply this to the functor

$$E \mapsto \operatorname{Hom}(\mathcal A(E),F)$$

where $\mathcal A : D^b( \operatorname{coh} X) \to D^b( \operatorname{coh} X^{an})$ is an analytification functor

[Matthew Ballard]

This produces an adjoint

[Adam Topaz]

scrap that... it should be more similar to something like a uniform structure on Arrow C.

[Matthew Ballard]

From looking at Neeman’s notes, I wonder how much a discussion of topology is getting in the way here