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 $\flat^{rel}$.
Where the plain flat modality $\flat$ sends any object $X$ to the type $\flat X$ of its global points, the relative flat modality instead sends it to the type $\flat^{rel} X$ of all infinitesimal disks (i.e. the infinitesimal neighbourhoods of all global points) in $X$.
See also at differential cohesion and idelic structure.
Given differential cohesion,
define operations $ʃ^{rel}$ and $\flat^{rel}$ by
Hence $ʃ^{rel} X$ makes a homotopy pushout square
and $\flat^{rel}$ makes a homotopy pullback square
We call $ʃ^{rel}$ the relative shape modality and $\flat^{rel}$ the relative flat modality.
The relative shape and flat modalities of def.
form an adjoint pair $(ʃ^{rel} \dashv \flat^{rel})$;
whose (co-)modal types are precisely the properly infinitesimal types, hence those for which $\flat \to \Im$ is an equivalence;
$ʃ^{rel}$ preserves the terminal object.
It follows that when $\flat^{rel}$ has a further right adjoint $\sharp^{rel}$ with equivalent modal types containing the codiscrete types, then this defines a level
hence an intermediate subtopos $\infty Grpd \hookrightarrow \mathbf{H}_{infinitesimal}\hookrightarrow \mathbf{H}_{th}$ 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 $(\flat^{rel} \dashv \sharp^{rel})$ does not provide Aufhebung for $(\flat \dashv \sharp)$.
(…)
The counit of the relative flat modality is a formally étale morphism.
Last revised on May 27, 2015 at 14:13:22. See the history of this page for a list of all contributions to it.