nLab
Pi-closed morphism

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

$$\begin{array}{ccc}X& \stackrel{{\eta }_X}{\to }& \Pi (X)\\ {\downarrow }^f& & {\downarrow }^{\Pi (f)}\\ Y& \stackrel{{\eta }_Y}{\to }& \Pi (Y)\end{array}$$\array{ X &\stackrel{\eta_X}{\to}& \mathbf{\Pi}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}(f)}} \\ Y &\stackrel{\eta_Y}{\to}& \mathbf{\Pi}(Y) }

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

Definition

Let $(\Pi ⊣Disc⊣\Gamma ):H\to \mathrm{\infty }Grpd$ be an infinity-connected (infinity,1)-topos, let $\Pi :=Disc\Pi$ be the geometric path functor / geometric homotopy functor, let $f:X\to Y$ be a $H$-morphism, let $c_{\Pi }f$ denote the ∞-pullback

$$\begin{array}{ccc}{c}_{\Pi }f& \to & \Pi X\\ \downarrow & & {\downarrow }^{{\Pi }_f}\\ Y& \stackrel{1_{(\Pi ⊣Disc)}}{\to }& \Pi Y\end{array}$$\array{c_{\mathbf{\Pi}} f&\to& {\mathbf{\Pi}} X\\\downarrow&&\downarrow^{{\mathbf{\Pi}}_f}\\Y&\xrightarrow{1_{(\Pi\dashv \Disc)}}&{\mathbf{\Pi}}Y}

$c_{\Pi }f$ is called $\Pi$-closure of $f$.

$f$ is called $\Pi$-closed if $X\simeq c_{\Pi }f$.

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 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.

Proposition

Let $H$ be a cohesive (∞,1)-topos possessing a ∞-cohesive site of definition. Then for $X\in H$ the locally constant ∞-stacks $E\in LConst(X)$, regarded as ∞-bundle morphisms $p:E\to X$ are precisely the $\Pi$-closed morphisms into $X$

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

• 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

category: cohesive (∞1)-topos