this is a subentry of cohesive (infinity,1)-topos. See there for background and context
We discuss how a cohesive (∞,1)-topos that is equipped with a notion of infinitesimal cohesion induces a notion of geometry (for structured (∞,1)-toposes), hence intrinsically defines a higher geometry with a good notion of cohesively structured (∞,1)-toposes that suitably adapts and generalizes the notion of locally ringed space and locally ringed toposes.
This means that for any (∞,1)-topos and
For this general abstract construction to indeed accurately model a notion of higher geometry, this setup needs to be equipped with a suitable choice of admissible morphisms between such -structure sheaves: not every morphis of classifying geometric morphisms qualifies as morphism of locally -algebra-ed -toposes. This extra datum is encoded by a choice of morphisms in that qualify as open maps in a suitable sense. Such a choice then gives rise to a genuine notion of geometry (for structured (∞,1)-toposes).
We discuss below how in the case that is a cohesive (∞,1)-topos equipped with infinitesimal cohesion? these open maps are canonically and intrinsically induced: they are the formally etale morphisms with respect to the given notion of infinitesimal cohesion.
Therefore we can give the following abstract characterization of local morphisms of “locally algebra-ed “-toposes (I’ll use the latter term – supposed to remind us that it generalizes the notion of locally ringed topos – tentatively for the moment, until I maybe settle for a better term). I would like to know if there is still nicer and way to think of the following.
So for our given cohesive -topos we regard it as the classifying -topos for some theory of local T-algebras. Then given any -topos a T-structure sheaf on is a geometric morphism
whose inverse image we write .
We then want to identify “étale” morphisms in and declare that a morphism of locally T-algebra-ed -toposes
is a geometric transformation as indicated, such that on étale morphisms in all its component naturality squares
are pullback squares.
In view of the above this looks like it might be a hint for a more powerful description: because the Rosenberg-Kontsevich characterization of the (formally) étale morphism is of the same, but converse form: given an infinitesimal cohesive neighbourhood
we have canonically given a natural transformation
and we say is (formally) étale if its comonents naturality squares under
So in total we are looking at diagrams of the form
and demand the compatibility condition that those morphisms in that have cartesian components under also have cartesian components under .