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 to the inverse image of the global sections geometric morphism induces a higher modality , 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 may be thought of as the shape of and therefore one may call the shape modality. It forms an adjoint modality with the flat modality .
Generally, given an (∞,1)-topos (or just a 1-topos) equipped with an idempotent monad (a (higher) modality/closure operator) which preserves (∞,1)-pullbacks over objects in its essential image, one may call a morphism in -closed if the unit-diagram
is an (∞,1)-pullback diagram. These -closed morphisms form the right half of an orthogonal factorization system, the left half being the morphisms that are sent to equivalences in .
Let be an infinity-connected (infinity,1)-topos, let be the geometric path functor / geometric homotopy functor, let be a -morphism, let denote the ∞-pullback
is called -closure of .
is called -closed if .
If a morphism factors into and is a -equivalence then is -closed; this is seen by using that is idempotent.
-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 -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 -stacks
In a cohesive (∞,1)-topos with an ∞-cohesive site of definition, the fundamental ∞-groupoid-functor satisfies the above assumptions (this is the example gives this entry its name). The -closed morphisms into some are canonically identified with the locally constant ∞-stacks over . The correspondence is effectively what is called categorical Galois theory.
Let be a cohesive (∞,1)-topos possessing a ∞-cohesive site of definition. Then for the locally constant ∞-stacks , regarded as ∞-bundle morphisms are precisely the -closed morphisms into
Formally étale morphisms
If a differential cohesive (∞,1)-topos , the de Rham space functor satisfies the above assumptions. The -closed morphisms are precisely the formally étale morphisms.
The examples of locally constant -stacks and of formally étale morphisms are discussed in sections 3.5.6 and 3.7.3 of