# nLab cohesive (infinity,1)-topos -- structure sheaves

this is a subentry of cohesive (infinity,1)-topos. See there for background and context

under construction

### 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

We discuss how a cohesive (∞,1)-topos that is equipped with a notion of infinitesimal cohesion induces a notion of geometry (for structured (∞,1)-toposes), hence intrinsically defines a higher geometry with a good notion of cohesively structured (∞,1)-toposes that suitably adapts and generalizes the notion of locally ringed space and locally ringed toposes.

Every (∞,1)-topos $H$ is (in a tautological but useful way), the classifying topos (see there for details) for a theory $𝕋$ of local ∞-algebras.

This means that for $𝒳$ any (∞,1)-topos and

$A:\left({𝒪}_{𝒳}⊣R\right):𝒳\stackrel{\stackrel{{𝒪}_{𝒳}}{←}}{\underset{R}{\to }}H$A : (\mathcal{O}_{\mathcal{X}} \dashv R) : \mathcal{X} \stackrel{\overset{\mathcal{O}_{\mathcal{X}}}{\leftarrow}}{\underset{R}{\to}} \mathbf{H}

a geometric morphism, we may think of the left exact and cover-preserving (hence “local”) functor

${𝒪}_{𝒳}:{𝒞}_{𝕋}\stackrel{j}{\to }H\stackrel{{𝒪}_{𝒳}}{\to }𝒳$\mathcal{O}_{\mathcal{X}} : \mathcal{C}_{\mathbb{T}} \stackrel{j}{\to} \mathbf{H} \stackrel{\mathcal{O}_{\mathcal{X}}}{\to} \mathcal{X}

given by the composition of the (∞,1)-Yoneda embedding of the syntactic site ${𝒞}_{𝕋}$ of $𝕋$ with the inverse image ${𝒪}_{𝒳}$ as exhibiting a structure sheaf of local $𝕋$-∞-algebras in $𝒳$.

For this general abstract construction to indeed accurately model a notion of higher geometry, this setup needs to be equipped with a suitable choice of admissible morphisms between such $\infty$-structure sheaves: not every morphis of classifying geometric morphisms qualifies as morphism of locally $𝕋$-algebra-ed $\left(\infty ,1\right)$-toposes. This extra datum is encoded by a choice of morphisms in $H$ that qualify as open maps in a suitable sense. Such a choice then gives rise to a genuine notion of geometry (for structured (∞,1)-toposes).

We discuss below how in the case that $H$ is a cohesive (∞,1)-topos equipped with infinitesimal cohesion these open maps are canonically and intrinsically induced: they are the formally etale morphisms with respect to the given notion of infinitesimal cohesion.

## Definition

### Locally structured $\infty$-toposes

Therefore we can give the following abstract characterization of local morphisms of “locally algebra-ed $\infty$“-toposes (I’ll use the latter term – supposed to remind us that it generalizes the notion of locally ringed topos – tentatively for the moment, until I maybe settle for a better term). I would like to know if there is still nicer and way to think of the following.

So for $H$ our given cohesive $\infty$-topos we regard it as the classifying $\infty$-topos for some theory of local T-algebras. Then given any $\infty$-topos $𝒳$ a T-structure sheaf on $𝒳$ is a geometric morphism

$A:𝒳\to H$A : \mathcal{X} \to \mathbf{H}

whose inverse image we write ${𝒪}_{X}$.

We then want to identify “étale” morphisms in $H$ and declare that a morphism of locally T-algebra-ed $\infty$-toposes $\left(f,\alpha \right):\left(𝒳,{𝒪}_{𝒳}\right)\to \left(𝒴,{𝒪}_{𝒴}\right)$

$\begin{array}{c}𝒳\\ ↑& {↖}^{{𝒪}_{𝒳}}\\ {}^{{f}^{*}}↑& {}^{\alpha }⇗& H\\ ↑& {↙}_{{𝒪}_{𝒴}}\\ 𝒴\end{array}$\array{ \mathcal{X} \\ \uparrow & \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}} \\ {}^{\mathllap{f^*}}\uparrow &{}^{\mathllap{\alpha}}\neArrow& \mathbf{H} \\ \uparrow & \swarrow_{\mathrlap{\mathcal{O}_{\mathcal{Y}}}} \\ \mathcal{Y} }

is a geometric transformation as indicated, such that on étale morphisms $p:Y\to X$ in $H$ all its component naturality squares

$\begin{array}{ccc}{f}^{*}{𝒪}_{𝒳}\left(Y\right)& \stackrel{{\alpha }_{Y}}{\to }& {𝒪}_{𝒴}\\ ↓& & ↓\\ {f}^{*}{𝒪}_{𝒳}\left(X\right)& \stackrel{{\alpha }_{X}}{\to }& {𝒪}_{𝒴}\end{array}$\array{ f^* \mathcal{O}_{\mathcal{X}}(Y) &\stackrel{\alpha_Y}{\to}& \mathcal{O}_{\mathcal{Y}} \\ \downarrow && \downarrow \\ f^* \mathcal{O}_{\mathcal{X}}(X) &\stackrel{\alpha_X}{\to}& \mathcal{O}_{\mathcal{Y}} }

are pullback squares.

In view of the above this looks like it might be a hint for a more powerful description: because the Rosenberg-Kontsevich characterization of the (formally) étale morphism $Y\to X$ is of the same, but converse form: given an infinitesimal cohesive neighbourhood

$i:H\to {H}_{\mathrm{th}}$i : \mathbf{H} \to \mathbf{H}_{\mathrm{th}}

we have canonically given a natural transformation

$\varphi :{i}_{!}⇒{i}_{*}$\phi : i_! \Rightarrow i_*

looking like

$\begin{array}{cc}& ↗{↘}^{{i}_{!}}\\ H& {⇓}^{\varphi }& {H}_{\mathrm{th}}\\ & ↘{↗}_{{i}_{*}}\end{array}$\array{ & \nearrow \searrow^{\mathrlap{i_!}} \\ \mathbf{H} & \Downarrow^{\phi}& \mathbf{H}_{th} \\ & \searrow \nearrow_{\mathrlap{i_*}} }

and we say $Y\to X$ is (formally) étale if its comonents naturality squares under $\varphi$

$\begin{array}{ccc}{i}_{!}X& \stackrel{{\varphi }_{Y}}{\to }& {i}_{!}Y\\ ↓& & ↓\\ {i}_{*}X& \stackrel{{\varphi }_{Y}}{\to }& {i}_{*}Y\end{array}$\array{ i_! X &\stackrel{\phi_Y}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{\phi_Y}{\to}& i_* Y }

are pullbacks.

So in total we are looking at diagrams of the form

$\begin{array}{c}𝒳\\ ↑& {↖}^{{𝒪}_{𝒳}}& & ↗{↘}^{{i}_{!}}\\ {}^{{f}^{*}}↑& {}^{\alpha }⇗& H& {⇓}^{\varphi }& {H}_{\mathrm{th}}\\ ↑& {↙}_{{𝒪}_{𝒴}}& & ↘{↗}_{{i}_{*}}\\ 𝒴\end{array}$\array{ \mathcal{X} \\ \uparrow & \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}} & & \nearrow \searrow^{\mathrlap{i_!}} \\ {}^{\mathllap{f^*}}\uparrow &{}^{\mathllap{\alpha}}\neArrow& \mathbf{H} &\Downarrow^{\phi}& \mathbf{H}_{th} \\ \uparrow & \swarrow_{\mathrlap{\mathcal{O}_{\mathcal{Y}}}} && \searrow \nearrow_{\mathrlap{i_*}} \\ \mathcal{Y} }

and demand the compatibility condition that those morphisms in $H$ that have cartesian components under $\varphi$ also have cartesian components under $\alpha$.

(…)

## References

Revised on March 5, 2012 23:42:01 by Tim Campion? (140.180.17.153)