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Am starting an entry pro-manifold. Have added statement and proof that pro-Cartesian spaces are fully faithful in smooth loci (here).
I am considering the category of towers of Cartesian spaces with pro-morphisms between them, i.e. the full subcategory
$TowCartSp \hookrightarrow Pro(CartSp)$of the pro-category of Cartesian spaces on those pro-objects that are formal sequential limits.
I think I wrote down a proof that equipped with “towers of good open covers”, this becomes a site:
The site of towers of Cartesian spaces and pro-morphisms.
But check. For the time being I have labeled the section as “under construction”.
So I am thinking it is clear that this site of towers of Cartesian spaces and pro-morphism is an “infinity-cohesive site”:
the conditions we need to check essentially follow from the fact that degreewise the towers of Cartesian spaces with towers of good open covers between them behave like in the infinity-cohesive site of plain Cartesian spaces
For instance use that coproducts commute with connected limits (and towers are connected) to deduce that the Cech nerve of a formal limit over a tower of good open covers is degreewise a coproduct of representables, etc.
I was in that kind of area during this musing.
I am giving this a section of its own, we should discuss at The pro-category of towers.
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