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Stream: deprecated: algebraic geometry

Topic: locally representable?

Matteo Capucci (he/him) (Jul 06 2022 at 08:30):

I just stumbled upon this sentence
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Matteo Capucci (he/him) (Jul 06 2022 at 08:30):

what's a locally representable functor? it doesn't seem to exist outside of that paragraph

Matteo Capucci (he/him) (Jul 06 2022 at 08:31):

(from here https://ncatlab.org/nlab/show/generalized+scheme)

Damiano Mazza (Jul 06 2022 at 09:00):

I don't know of a general definition outside of algebraic-geometry-related contexts, but Definition 2.2 in these notes by Bertrand Toën is a possible axiomatization beyond the case of schemes. The particular case of schemes is described, independently of Toën's axiomatization, in the related nLab article, in the section "As sheaves on $\mathbf{CRing}^\mathrm{op}$". Although the phrase "locally representable sheaf" is never used, it corresponds to point (2) of Definition 2.4 on that page.

Essentially, Toën's approach starts with defining what is an "open embedding" of sheaves on some site. The notion of local representability is parametric in the class of open embeddings: a sheaf $X$ is "locally representable" if there is a family of open embeddings $\{U_i\to X\}_{i\in I}$ where each $U_i$ is a representable sheaf, such that $\sum_{i\in I} U_i\to X$ is an epimorphism.

Also, I haven't read it in detail, but I think that @Zhen Lin Low's Ph.D thesis is very relevant here.

Zhen Lin Low (Jul 06 2022 at 10:13):

My thesis is a proposed general definition... but, really, it is not a well-defined term. In a sense, any presheaf whatsoever is locally representable.

Morgan Rogers (he/him) (Jul 06 2022 at 10:21):

I think a meaningful distinction could be made sometimes between arbitrary presheaves and those which are covered by representable sub-presheaves subject to some restriction. That distinction breaks down as soon as your representables are closed under quotients in the topos though.

Zhen Lin Low (Jul 06 2022 at 10:33):

There's some easy stuff about disjoint unions (i.e. lextensive coproducts) too, but for simplicity we may assume that disjoint unions of representable sheaves are representable. (Disjoint unions of representable presheaves tend not to be representable, though.) Then yes, it basically amounts to constructing exact quotients of representables by nice equivalence relations. The curious thing is that niceness condition is not itself nice...

Damiano Mazza (Jul 06 2022 at 12:11):

Zhen Lin Low said:

My thesis is a proposed general definition... but, really, it is not a well-defined term. In a sense, any presheaf whatsoever is locally representable.

Are you alluding to the fact that, if you're too liberal with how you define an "open cover" of presheaves, the fact that every presheaf is a colimit of representables ends up making every presheaf locally representable? (I have the feeling that you and @Morgan Rogers (he/him) are talking about something more subtle, but I don't know...).

Matteo Capucci (he/him) (Jul 06 2022 at 14:46):

I see, thanks!