structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In a locally ∞-connected (∞,1)-topos with fully faithfiil 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 $\int \coloneqq Disc \circ \Pi$, which sends an object to something that may be regarded equivalently as its geometric realization or its fundamental ∞-groupoid. In either case $\int X$ may be thought of as the shape of $X$ and therefore one may call $\int$ 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 $\mathbf{\Pi} \colon \mathbf{H} \to \mathbf{H}$ (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 $\mathbf{H}$ $\mathbf{\Pi}$-closed if the unit-diagram
is an (∞,1)-pullback diagram. These $\mathbf{\Pi}$-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 ${\mathbf{\Pi}}:=\Disc \Pi$ be the geometric path functor / geometric homotopy functor, let $f:X\to Y$ be a $H$-morphism, let $c_{\mathbf{\Pi}} f$ denote the ∞-pullback
$c_{\mathbf{\Pi}} f$ is called ${\mathbf{\Pi}}$-closure of $f$.
$f$ is called ${\mathbf{\Pi}}$-closed if $X\simeq c_{\mathbf{\Pi}}f$.
If a morphism $f:X\to Y$ factors into $f=g\circ h$ and $h$ is a $\mathbf{\Pi}$-equivalence then $g$ is $\mathbf{\Pi}$-closed; this is seen by using that ${\mathbf{\Pi}}$ 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 $\mathbf{\Pi}$-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 $\mathbf{\Pi}$ satisfies the above assumptions (this is the example gives this entry its name). The $\mathbf{\Pi}$-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 $\mathbf{\Pi}$-closed morphisms into $X$
If a differential cohesive (∞,1)-topos $\mathbf{H}_{th}$, the de Rham space functor $\mathbf{\Pi}_{inf}$ satisfies the above assumptions. The $\mathbf{\Pi}_{inf}$-closed morphisms are precisely the formally étale morphisms.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(Red \dashv ʃ_{inf} \dashv \flat_{inf})$
The examples of locally constant $\infty$-stacks and of formally étale morphisms are discussed in sections 3.5.6 and 3.7.3 of