Contents

Idea

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 $\esh \coloneqq Disc \circ \Pi$, 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 and at shape via cohesive path ∞-groupoid). In either case $\esh X$ may be thought of as the shape of $X$ and therefore one may call $\esh$ the shape modality. It forms an adjoint modality with the flat modality $\flat \coloneqq Disc \circ \Gamma$.

Properties

Relative shape and factorization system

Generally, given an (∞,1)-topos $\mathbf{H}$ (or just a 1-topos) equipped with an idempotent monad $\esh \colon \mathbf{H} \to \mathbf{H}$ (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 $\mathbf{H}$ $\esh$-closed if the unit-diagram

$\array{ X &\stackrel{\eta_X}{\to}& \esh(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\esh(f)}} \\ Y &\stackrel{\eta_Y}{\to}& \esh(Y) }$

is an (∞,1)-pullback diagram. These $\esh$-closed morphisms form the right half of an orthogonal factorization system, the left half being the morphisms that are sent to equivalences in $\mathbf{H}$.

Definition

Let $(\Pi\dashv \Disc\dashv \Gamma):H\to\infty\Grpd$ be an infinity-connected (infinity,1)-topos, let $\esh:=\Disc \Pi$ be the geometric path functor / geometric homotopy functor, let $f:X\to Y$ be a $H$-morphism, let $c_{\esh} f$ denote the ∞-pullback

$\array{c_{\esh} f&\to& {\esh} X\\\downarrow&&\downarrow^{{\esh}_f}\\Y&\xrightarrow{1_{(\Pi\dashv \Disc)}}&{\esh}Y}$

$c_{\esh} f$ is called $\esh$-closure of $f$.

$f$ is called $\esh$-closed if $X\simeq c_{\esh}f$.

If a morphism $f:X\to Y$ factors into $f=g\circ h$ and $h$ is a $\esh$-equivalence then $g$ is $\esh$-closed; this is seen by using that $\esh$ 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.

As open maps

A consequence of the previous property is that the class of $\esh$-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.

Examples

Internal locally constant $\infty$-stacks

In a cohesive (∞,1)-topos $\mathbf{H}$ with an ∞-cohesive site of definition, the fundamental ∞-groupoid-functor $\esh$ satisfies the above assumptions (this is the example gives this entry its name). The $\esh$-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.

Proposition

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 $\esh$-closed morphisms into $X$

Formally étale morphisms

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.

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

References

In Smooth∞Grpd

For the case of Smooth∞Grpd:

Discussion for orbifolds, étale groupoids and, generally, étale ∞-groupoids: