# nLab shape modality

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $\left(\infty ,1\right)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion

# 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

$\begin{array}{ccc}X& \stackrel{{\eta }_{X}}{\to }& \Pi \left(X\right)\\ {↓}^{f}& & {↓}^{\Pi \left(f\right)}\\ Y& \stackrel{{\eta }_{Y}}{\to }& \Pi \left(Y\right)\end{array}$\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 $\left(\Pi ⊣Disc⊣\Gamma \right):H\to \infty Grpd$ 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

$\begin{array}{ccc}{c}_{\Pi }f& \to & \Pi X\\ ↓& & {↓}^{{\Pi }_{f}}\\ Y& \stackrel{{1}_{\left(\Pi ⊣Disc\right)}}{\to }& \Pi Y\end{array}$\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 a kind of open maps.

## Examples

### Internal locally constant $\infty$-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 $E\in LConst\left(X\right)$, 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 ${\Pi }_{\mathrm{inf}}$ satisfies the above assumptions. The ${\Pi }_{\mathrm{inf}}$-closed morphisms are precisely the formally étale morphisms.

cohesion

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

$\left(ʃ⊣♭⊣♯\right)$

differential cohesion

## References

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

Revised on January 5, 2013 21:44:54 by Urs Schreiber (89.204.138.93)