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Tim Hosgood (Apr 09 2020 at 16:51):(cross-posted from here (sorry) because i didn't realise that this stream existed!)
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"
(some definitions, for clarity)
Screenshot-2020-04-07-at-02.35.23.png Screenshot-2020-04-07-at-02.35.13.png
