structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In a locally ∞-connected (∞,1)-topos with fully faithful inverse image (such as a cohesive (∞,1)-topos), the extra left adjoint $\Pi$ to the inverse image $Disc$ of the global sections geometric morphism $\Gamma$ induces a higher modality $ʃ \coloneqq Disc \circ \Pi$, which sends an object to something that may be regarded equivalently as its geometric realization or its fundamental ∞-groupoid (see at fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos). In either case $ʃ X$ may be thought of as the shape of $X$ and therefore one may call $ʃ$ the shape modality. It forms an adjoint modality with the flat modality $\flat \coloneqq Disc \circ \Gamma$.
Generally, given an (∞,1)-topos $\mathbf{H}$ (or just a 1-topos) equipped with an idempotent monad $ʃ \colon \mathbf{H} \to \mathbf{H}$ (a (higher) modality/closure operator) which preserves (∞,1)-pullbacks over objects in its essential image, one may call a morphism $f \colon X \to Y$ in $\mathbf{H}$ $ʃ$-closed if the unit-diagram
is an (∞,1)-pullback diagram. These $ʃ$-closed morphisms form the right half of an orthogonal factorization system, the left half being the morphisms that are sent to equivalences in $\mathbf{H}$.
Let $(\Pi\dashv \Disc\dashv \Gamma):H\to\infty\Grpd$ be an infinity-connected (infinity,1)-topos, let $ʃ:=\Disc \Pi$ be the geometric path functor / geometric homotopy functor, let $f:X\to Y$ be a $H$-morphism, let $c_{ʃ} f$ denote the ∞-pullback
$c_{ʃ} f$ is called $ʃ$-closure of $f$.
$f$ is called $ʃ$-closed if $X\simeq c_{ʃ}f$.
If a morphism $f:X\to Y$ factors into $f=g\circ h$ and $h$ is a $ʃ$-equivalence then $g$ is $ʃ$-closed; this is seen by using that $ʃ$ is idempotent.
$\Pi$-closed morphisms are a right class of an orthogonal factorization system (in an (∞,1)-category) and hence, as discussed there, are closed under limits, composition, retracts and satisfy the left cancellation property.
A consequence of the previous property is that the class of $ʃ$-closed morphisms gives rise to an admissible structure in the sense of structured spaces on an (∞,1)-connected (∞,1)-topos, hence they serve as a class of a kind of open maps.
In a cohesive (∞,1)-topos $\mathbf{H}$ with an ∞-cohesive site of definition, the fundamental ∞-groupoid-functor $ʃ$ satisfies the above assumptions (this is the example gives this entry its name). The $ʃ$-closed morphisms into some $X \in \mathbf{H}$ are canonically identified with the locally constant ∞-stacks over $X$. The correspondence is effectively what is called categorical Galois theory.
Let $H$ be a cohesive (∞,1)-topos possessing a ∞-cohesive site of definition. Then for $X\in H$ the locally constant ∞-stacks $E\in \L\Const(X)$, regarded as ∞-bundle morphisms $p:E\to X$ are precisely the $ʃ$-closed morphisms into $X$
If a differential cohesive (∞,1)-topos $\mathbf{H}_{th}$, the de Rham space functor $\Im$ satisfies the above assumptions. The $\Im$-closed morphisms are precisely the formally étale morphisms.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
The examples of locally constant $\infty$-stacks and of formally étale morphisms are discussed in sections 3.5.6 and 3.7.3 of
See also
for further discussion of the smooth shape modality of cohesion (the etale homotopy type operation in the context of smooth infinity-stacks) as applied to orbifolds and étale groupoids and generally étale ∞-groupoids.