shape modality


Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory



Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?


Modalities, Closure and Reflection



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 ʃDiscΠʃ \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 ʃXʃ X may be thought of as the shape of XX and therefore one may call ʃʃ the shape modality. It forms an adjoint modality with the flat modality DiscΓ\flat \coloneqq Disc \circ \Gamma.


Relative shape and factorization system

Generally, given an (∞,1)-topos H\mathbf{H} (or just a 1-topos) equipped with an idempotent monad ʃ: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} ʃʃ-closed if the unit-diagram

X η X ʃ(X) f ʃ(f) Y η Y ʃ(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 ʃʃ-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}.


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

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

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

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

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


Internal locally constant \infty-stacks

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


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 ʃʃ-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.


tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale contractible ʃ discrete discrete differential * \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{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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 April 14, 2015 14:50:32 by Urs Schreiber (