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 $\flat \coloneqq Disc \circ \Gamma$.
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
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 $\mathbf{H}$.
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
$c_{\esh} f$ 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.
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.
See at shape via cohesive path ∞-groupoid.
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.
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 $X$
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.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Urs Schreiber, Sec. 3.4.5 in: differential cohomology in a cohesive topos (arXiv:1310.7930)
Mike Shulman, Sec. 9.7 in: Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)
For the case of Smooth∞Grpd:
On shape via cohesive path ∞-groupoids:
Dmitri Pavlov, Structured Brown representability via concordance, 2014 (pdf, pdf)
Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov, Classifying spaces of infinity-sheaves (arXiv:1912.10544)
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.