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Given a site $\mathcal{C}$ (with possibly extra structure), is there a good notion of a "completion" $\overline{\mathcal{C}}$ of $\mathcal{C}$ which results in a site of things that "locally look like" objects of $\mathcal{C}$? Specifically, in nice situations it would be ideal if anything resembling the following (especially the equivalence between 4+5 and 6) hold:

  1. $\overline{\mathcal{C}}$ is a site,
  2. $\mathcal{C}\subseteq\overline{\mathcal{C}}$ is a full subcategory with no more covers than $\mathcal{C}$ originally had,
  3. $\overline{\overline{\mathcal{C}}}=\overline{\mathcal{C}}$,
  4. An object of $\mathcal{C}$ can be given by a specification of Čech/torsor datum, i.e. collections of objects $\left\{X_i\right\}$, "open" subobjects $\left\{U_{ijk}\subseteq X_i\right\}$ such that the allowed indices $k$ depend only on the unordered pair $\{i,j\}$, and isomorphisms $\varphi_{ijk}:U_{ijk}\rightarrow U_{jik}$ satisfying a cocycle condition,
  5. Every object $X$ has at least one specification in terms of this sort of datum, and for any such specification the objects $\{X_i\}$ cover $X$ in $\overline{\mathcal{C}}$,
  6. An object of $\mathcal{C}$ is also the same as a sheaf on $\mathcal{C}$ admitting a "cover" by representable sheaves, and this is a covering in $\overline{\mathcal{C}}$,
  7. $\operatorname{Sh}\left(\overline{\mathcal{C}}\right)=\operatorname{Sh}(\mathcal{C})$,
  8. If $X$ is a scheme and $\mathcal{C}$ is the Zariski site of (opposite) quasicoherent $\mathcal{O}_X$-algebras, then $\overline{\mathcal{C}}=\mathbf{Sch}/X$,
  9. If $k\in\mathbb{N}\cup\{\infty\}$, $\mathcal{C}$ is the site with one object $\mathbb{R}^n$, $C^k$ maps as morphisms, and topological open coverings by injective local diffeomorphisms as covers (or some site of this sort), then $\overline{\mathcal{C}}$ should be $n$-dimensional $C^k$-manifolds,
  10. If $\mathcal{C}$ has the indiscrete topology, then $\overline{\mathcal{C}}=\operatorname{Sh}\mathcal{C}=\operatorname{PSh}\mathcal{C}$ via the Yoneda embedding. Thus, for example, if $\Delta$ is the simplicial category then $\overline\Delta$ is the category of simplicial sets.

I can find many references making the functor formalism 6 for this precise in special cases of #8 that should easily generalize to all of #8, but I can find no reference which tries to make the construction of such completions systematic, or abstractly describe the equivalence with the Čech viewpoint. (In the scheme case, the equivalence between 4+5 and 6 basically consists of a standard representability lemma for schemes, e.g. Stacks 26.15.4.) Criterion 7 also makes me wonder if this can be thought of as some adjoint to the functor $\operatorname{Sh}$ from sites to (Grothendieck) topoi. For reference, I know essentially nothing about topoi.

For convenience, I am pretending there is no such thing as a "set-theoretic issue."

Curious
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  • Out of curiosity, what are some examples of this aside from manifolds (which locally look like $\mathbb R^k$)? – littleO Mar 11 '21 at 18:10
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    @littleO We have silly abstract examples like "local Lipschitz structures" (complete the category of metric spaces with Lipschitz maps), maybe variations on this? We have algebraic examples like Zariski/étale/flat topologies on $\mathbf{Ring}^{\mathrm{op}}$ all giving the same categories of schemes. We also have variations on manifolds like orbifolds, classes of complex or almost-complex manifolds, or analytic manifolds over a complete DVR, or rigid analytic spaces. If C is the simplicial category with the indiscrete topology, sheaves (hopefully $\overline{\mathcal{C}}$) are simplicial sets. – Curious Mar 11 '21 at 18:26
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    @littleO Fiber bundles, as well as spaces equipped with fiber bundles, are also examples (thus why I call 4 "Čech/torsor datum"). Nothing else is immediately coming to mind. In light of the simplicial example, I'll toss in an additional thing to hope for. – Curious Mar 11 '21 at 18:29
  • When you say that $\mathcal{C}$ is a site do you mean a category $\mathcal{C}$ (with pullbacks) equipped with a Grothendieck pretopology $\tau$ or do you mean a category $\mathcal{C}$ (not necessarily with pullbacks) equipped with a Grothendieck topology $J$? – Geoff Mar 11 '21 at 19:19
  • @Geoff The category of $C^k$ manifolds will not have fibered products in general, so ideally I mean a topology on a category without pullbacks. I had been confusing this point slightly; therefore I suppose when I say "injective local diffeomorphisms" I should say "local diffeomorphisms," and maybe require all morphisms in the $\mathbb{R}^n$ category to be local diffeomorphisms. But any reasonable amount of fudging this point ought to be fine, since either way you get a site. – Curious Mar 11 '21 at 19:33
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    Zhen Lin's thesis is aimed at precisely these problems. It doesn't match your proposed framework perfectly but perhaps you'd like to have a look. https://www.repository.cam.ac.uk/bitstream/handle/1810/256998/Low-2016-PhD.pdf?sequence=1&isAllowed=y – Kevin Carlson Mar 11 '21 at 19:42
  • I'm becoming more convinced that criterion 10 shouldn't hold as written, this is a coincidence having to do with the fact that the indiscrete and canonical topologies on $\Delta$ coincide (https://math.stackexchange.com/questions/1633230/a-grothendieck-topology-on-delta). – Curious Mar 11 '21 at 21:01
  • My thesis is an attempt at a general framework for manifolds, schemes, etc. One thing I found difficult to avoid is the need to take "local homeomorphism" (or "étale morphism" or "open embedding" or whatever) as an additional datum, above and beyond the definition of "covering". I spent many months trying to find a way to define "open embedding" using categorical intrinsics without success. – Zhen Lin Mar 11 '21 at 23:51

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