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On the nLab, given a local $S$-topos $E$, a concrete sheaf is defined as an object that is separated with respect to the local isomorphisms (the morphisms that are inverted by the global sections functor $\Gamma:E\rightarrow S$): https://ncatlab.org/nlab/show/concrete+sheaf#in_a_local_topos

However, separated means that these morphisms are merely send to monos and not to isos. So I cannot see why a concrete sheaf would, in particular, be a sheaf.

I'm pretty sure I made a mistake in my reasoning, so I was wondering if somebody had a simple explanation.

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  • $\begingroup$ Given that objects of $E$ are sheaves, the V-separated objects in E are V-separated sheaves, i.e., concrete sheaves. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 28, 2022 at 13:41
  • $\begingroup$ Oh I think I see where I messed up. Sheaves are local objects with respect to the geometric morphism to the category of presheaves (on some site), which has nothing to do with the morphisms in the definition of concrete sheaves. So they are local with respect to one embedding and separated with respect to the geometric morphism $E\rightarrow S$? $\endgroup$
    – NDewolf
    Commented Sep 28, 2022 at 13:56
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    $\begingroup$ Yes, there are two entirely different Grothendieck topologies being used here. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 28, 2022 at 14:35
  • $\begingroup$ As a follow-up question. Does it give something extra to require the concrete sheaves to also be $V$-local? Hence, have some kind of double sheaf structure. $\endgroup$
    – NDewolf
    Commented Sep 28, 2022 at 14:38
  • $\begingroup$ Requiring V-locality makes the presheaf F trivial: its value on an object X is given by F(*)^{X(*)}, where * denotes the terminal object in the site, and F(*) denotes the set of points of F. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 28, 2022 at 14:50

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To summarize the discussion in the comments:

  • There are two nontrivial Grothendieck topologies used in the definition of a concrete sheaf: the Grothendieck topology T used to define sheaves, and the Grothendieck topology C used to define concrete presheaves.

  • The Grothendieck topology T (and its site S) are given to us.

  • The Grothendieck topology C is defined by specifying the generating covering family of an object X∈S as the family of all maps 1→X, where 1 is the terminal object in the site S.

  • C-separated presheaves are precisely concrete presheaves on S in the usual sense.

  • In particular, C-separated T-local presheaves on S (i.e., C-separated T-sheaves) are precisely concrete sheaves on S.

  • A C-local presheaf F (i.e., a C-sheaf) has a very simple form: it sends an object Y∈S to the set of maps Y(*)→F(*), where F(*) denotes the set of maps 1→F, where 1 is the terminal object of S.

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  • $\begingroup$ Many thanks for this clear overview! $\endgroup$
    – NDewolf
    Commented Sep 30, 2022 at 10:04

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