nLab
shape modality

Contents

Context

Cohesive \infty-Toposes

Modalities, Closure and Reflection

Contents

Idea

In a locally ∞-connected (∞,1)-topos with fully faithful inverse image (such as a cohesive (∞,1)-topos), the extra left adjoint Π\Pi to the inverse image DiscDisc of the global sections geometric morphism Γ\Gamma induces a higher modality ʃDiscΠ\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 ʃX\esh X may be thought of as the shape of XX and therefore one may call ʃ\esh the shape modality. It forms an adjoint modality with the flat modality DiscΓ\flat \coloneqq Disc \circ \Gamma.

Properties

Relative shape and factorization system

Generally, given an (∞,1)-topos H\mathbf{H} (or just a 1-topos) equipped with an idempotent monad ʃ:HH\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:XYf \colon X \to Y in H\mathbf{H} ʃ\esh-closed if the unit-diagram

X η X ʃ(X) f ʃ(f) Y η Y ʃ(Y) \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 ʃ\esh-closed morphisms form the right half of an orthogonal factorization system, the left half being the morphisms that are sent to equivalences in H\mathbf{H}.

Definition

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

c ʃf ʃX ʃ f Y 1 (ΠDisc) ʃY\array{c_{\esh} f&\to& {\esh} X\\\downarrow&&\downarrow^{{\esh}_f}\\Y&\xrightarrow{1_{(\Pi\dashv \Disc)}}&{\esh}Y}

c ʃfc_{\esh} f is called ʃ\esh-closure of ff.

ff is called ʃ\esh-closed if Xc ʃfX\simeq c_{\esh}f.

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

Π\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 ʃ\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.

Shape via cohesive path ∞-groupoid

See at shape via cohesive path ∞-groupoid.

Examples

Internal locally constant \infty-stacks

In a cohesive (∞,1)-topos H\mathbf{H} with an ∞-cohesive site of definition, the fundamental ∞-groupoid-functor ʃ\esh satisfies the above assumptions (this is the example gives this entry its name). The ʃ\esh-closed morphisms into some XHX \in \mathbf{H} are canonically identified with the locally constant ∞-stacks over XX. The correspondence is effectively what is called categorical Galois theory.

Proposition

Let HH be a cohesive (∞,1)-topos possessing a ∞-cohesive site of definition. Then for XHX\in H the locally constant ∞-stacks ELConst(X)E\in \L\Const(X), regarded as ∞-bundle morphisms p:EXp:E\to X are precisely the ʃ\esh-closed morphisms into XX

Formally étale morphisms

If a differential cohesive (∞,1)-topos H th\mathbf{H}_{th}, the de Rham space functor \Im satisfies the above assumptions. The \Im-closed morphisms are precisely the formally étale morphisms.

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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

General

In Smooth∞Grpd

For the case of Smooth∞Grpd:

On shape via cohesive path ∞-groupoids:

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

Last revised on October 4, 2021 at 04:08:22. See the history of this page for a list of all contributions to it.