Contents

Idea

Given an (∞,1)-topos $H$ (or just a 1-topos) equipped with an idempotent monad $\Pi :H\to H$ (a (higher) modality/closure operator) which preserves (∞,1)-pullbacks over objects in its essential image, one may call a morphism $f:X\to Y$ in $H$ $\Pi$-closed if the unit-diagram

is an (∞,1)-pullback diagram. These $\Pi$-closed morphisms form the right half of an orthogonal factorization system, the left half being the morphisms that are sent to equivalences in $H$.

Definition

If a morphism $f:X\to Y$ factors into $f=g\circ h$ and $h$ is a $\Pi$-equivalence then $g$ is $\Pi$-closed; this is seen by using that $\Pi$ is idempotent.

Properties

General

$\Pi$-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 $\Pi$-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 $\mathrm{\infty }$-stacks

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

Formally étale morphisms

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

Related entries

cohesion

• (shape modality $⊣$ flat modality $⊣$ sharp modality)

  $(ʃ⊣♭⊣♯)$

  • discrete object, codiscrete object, concrete object

  • structures in cohesion

differential cohesion

• (reduction modality $⊣$ infinitesimal shape modality $⊣$ infinitesimal flat modality)

  $(\mathrm{Red}⊣{ʃ}_{\inf }⊣{♭}_{\inf })$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

• differential cohesion - cotangent bundles

References

The examples of locally constant $\mathrm{\infty }$-stacks and of formally étale morphisms are discussed in sections 3.5.6 and 3.7.3 of

• Urs Schreiber, differential cohomology in a cohesive topos