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created concrete site
This definition reproduces the definition from Version 2 of https://arxiv.org/abs/0807.1704.
In Version 3, Baez and Hoffnung added the requirement that the site is subcanonical, and even mentioned this correction in the arXiv comment:
43 pages, version to be published; includes corrected definition of “concrete site”
Why is this requirement necessary? And if it is, we should probably add it to the article.
Perhaps just because they want the representables to be concrete sheaves (Prop 29)?
…which is used in the proofs of their Props 45 (the category of concrete sheaves is lcc) and 51 (every concrete sheaf is a colimit of representables).
It is definitely is referenced in Propositions 45 and 51, but is it actually necessary to prove them?
Johnstone (Elephant, Theorem C2.2.13) proves that $k$-separated objects in a Grothendieck topos (for some local operator $k$) form a locally presentable, locally cartesian closed, quasi-effective, cocomplete quasitopos with a generating set.
This takes care of Proposition 45, in particular, and does not require the site to be subcanonical.
As for Proposition 51, can we not replace representables by their sheafifications?
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