nLab
shape modality

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

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 Unknown characterUnknown characterDiscΠʃ \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 Unknown characterUnknown characterXʃ X may be thought of as the shape of XX and therefore one may call Unknown characterUnknown characterʃ 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 Unknown characterUnknown character:HHʃ \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} Unknown characterUnknown characterʃ-closed if the unit-diagram

X η X Unknown characterUnknown character(X) f Unknown characterUnknown character(f) Y η Y Unknown characterUnknown character(Y) \array{ X &\stackrel{\eta_X}{\to}& ʃ(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{ʃ(f)}} \\ Y &\stackrel{\eta_Y}{\to}& ʃ(Y) }

is an (∞,1)-pullback diagram. These Unknown characterUnknown characterʃ-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 Unknown characterUnknown character:=DiscΠʃ:=\Disc \Pi be the geometric path functor / geometric homotopy functor, let f:XYf:X\to Y be a HH-morphism, let c Unknown characterUnknown characterfc_{ʃ} f denote the ∞-pullback

c Unknown characterUnknown characterf Unknown characterUnknown characterX Unknown characterUnknown character f Y 1 (ΠDisc) Unknown characterUnknown characterY\array{c_{ʃ} f&\to& {ʃ} X\\\downarrow&&\downarrow^{{ʃ}_f}\\Y&\xrightarrow{1_{(\Pi\dashv \Disc)}}&{ʃ}Y}

c Unknown characterUnknown characterfc_{ʃ} f is called Unknown characterUnknown characterʃ-closure of ff.

ff is called Unknown characterUnknown characterʃ-closed if Xc Unknown characterUnknown characterfX\simeq c_{ʃ}f.

If a morphism f:XYf:X\to Y factors into f=ghf=g\circ h and hh is a Unknown characterUnknown characterʃ-equivalence then gg is Unknown characterUnknown characterʃ-closed; this is seen by using that Unknown characterUnknown characterʃ 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 Unknown characterUnknown characterʃ-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 H\mathbf{H} with an ∞-cohesive site of definition, the fundamental ∞-groupoid-functor Unknown characterUnknown characterʃ satisfies the above assumptions (this is the example gives this entry its name). The Unknown characterUnknown characterʃ-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 Unknown characterUnknown characterʃ-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

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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

References

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.

Revised on July 3, 2017 07:49:49 by Urs Schreiber (88.77.226.246)