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In a context of differential cohesion a reduction modality exhibits an inclusion of its modal types – the reduced objects. Essentially the corresponding inclusion of the anti-modal types is exhibited by an induced modal operator, the relative flat modality .
Where the plain flat modality sends any object to the type of its global points, the relative flat modality instead sends it to the type of all infinitesimal disks (i.e. the infinitesimal neighbourhoods of all global points) in .
See also at differential cohesion and idelic structure.
Given differential cohesion,
define operations and by
Hence makes a homotopy pushout square
and makes a homotopy pullback square
We call the relative shape modality and the relative flat modality.
The relative shape and flat modalities of def.
form an adjoint pair ;
whose (co-)modal types are precisely the properly infinitesimal types, hence those for which is an equivalence;
preserves the terminal object.
It follows that when has a further right adjoint with equivalent modal types containing the codiscrete types, then this defines a level
hence an intermediate subtopos which is infinitesimally cohesive.
This happens notably for the model of formal smooth ∞-groupoids and all its variants such as formal complex analytic ∞-groupoids etc. But in this case does not provide Aufhebung for .
(…)
The counit of the relative flat modality is a formally étale morphism.
Last revised on May 27, 2015 at 18:13:22. See the history of this page for a list of all contributions to it.