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Tim Hosgood (Apr 07 2020 at 00:34):In a few papers by Toledo and Tong, as well as a thesis by Green, the term "simplicial sheaf" or "simplicial vector bundle" is used in a way that is very different from e.g. the nLab's definition of the phrase. Indeed, these "simplicial sheaves" are really (co)lax homotopy limits of sheaves on the Čech nerve, and so are not just simplicial objects in some category of sheaves (namely because, in each simplicial degree, the sheaf lives over a different space (the corresponding degree of the Čech nerve), and also because they are really cosimplicial objects anyhow). Is there a more common name for these objects? The only thing I've seen them called is "one-cocycles of compatible isomorphisms" (in another Toledo & Tong paper). They turn out to be super useful* but it's hard to find out more about them without knowing a name.
*depending on your definitions of "super" and "useful"
Tim Hosgood (Apr 07 2020 at 00:35):(some definitions, for clarity)
Screenshot-2020-04-07-at-02.35.23.png Screenshot-2020-04-07-at-02.35.13.png
Tim Hosgood (Apr 08 2020 at 19:22):i’d also be interested in finding out how these things relate to actual simplicial sheaves (if at all)
sarahzrf (Apr 08 2020 at 20:08):tim posting again in "basic questions" about lax homotopy limits :pensive:
Tim Hosgood (Apr 08 2020 at 20:24):(sorry! but in my defence, all of the comp-sci stuff in general questions is also wayyyy over my head)
Tim Hosgood (Apr 08 2020 at 20:25):(also, just to clarify, i’m mostly just asking for some clarification on nomenclature (but obviously any extra info is an absolute bonus))
Tim Hosgood (Apr 08 2020 at 20:26):might move this to the alg geom stream though (if that’s possible?)
sarahzrf (Apr 08 2020 at 20:48):mostly just poking fun
Morgan Rogers (he/him) (Apr 09 2020 at 08:43):Tim Hosgood said:
might move this to the alg geom stream though (if that’s possible?)
There's currently no way to do this; just open a topic over there about it :grinning_face_with_smiling_eyes:
Tim Hosgood (Apr 09 2020 at 16:52):Morgan Rogers said:
Tim Hosgood said:
might move this to the alg geom stream though (if that’s possible?)
There's currently no way to do this; just open a topic over there about it :grinning_face_with_smiling_eyes:
just did that here, thanks!