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

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

under construction

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 $\mathbf{H}$ is (in a tautological but useful way), the classifying topos (see there for details) for a theory $\mathbb{T}$ of local ∞-algebras.

This means that for $\mathcal{X}$ any (∞,1)-topos and

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

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

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

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

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 $\mathbb{T}$-algebra-ed $(\infty,1)$-toposes. This extra datum is encoded by a choice of morphisms in $\mathbf{H}$ 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 $\mathbf{H}$ 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 $\mathbf{H}$ 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 $\mathcal{X}$ a T-structure sheaf on $\mathcal{X}$ is a geometric morphism

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

whose inverse image we write $\mathcal{O}_X$.

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

$$\array{ \mathcal{X} \\ \uparrow & \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}} \\ {}^{\mathllap{f^*}}\uparrow &{}^{\mathllap{\alpha}}\neArrow& \mathbf{H} \\ \uparrow & \swarrow_{\mathrlap{\mathcal{O}_{\mathcal{Y}}}} \\ \mathcal{Y} }$$

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

$$\array{ f^* \mathcal{O}_{\mathcal{X}}(Y) &\stackrel{\alpha_Y}{\to}& \mathcal{O}_{\mathcal{Y}} \\ \downarrow && \downarrow \\ f^* \mathcal{O}_{\mathcal{X}}(X) &\stackrel{\alpha_X}{\to}& \mathcal{O}_{\mathcal{Y}} }$$

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

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

we have canonically given a natural transformation

$$\phi : i_! \Rightarrow i_*$$

looking like

$$\array{ & \nearrow \searrow^{\mathrlap{i_!}} \\ \mathbf{H} & \Downarrow^{\phi}& \mathbf{H}_{th} \\ & \searrow \nearrow_{\mathrlap{i_*}} }$$

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

$$\array{ i_! X &\stackrel{\phi_Y}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{\phi_Y}{\to}& i_* Y }$$

are pullbacks.

So in total we are looking at diagrams of the form

$$\array{ \mathcal{X} \\ \uparrow & \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}} & & \nearrow \searrow^{\mathrlap{i_!}} \\ {}^{\mathllap{f^*}}\uparrow &{}^{\mathllap{\alpha}}\neArrow& \mathbf{H} &\Downarrow^{\phi}& \mathbf{H}_{th} \\ \uparrow & \swarrow_{\mathrlap{\mathcal{O}_{\mathcal{Y}}}} && \searrow \nearrow_{\mathrlap{i_*}} \\ \mathcal{Y} }$$

and demand the compatibility condition that those morphisms in $\mathbf{H}$ that have cartesian components under $\phi$ also have cartesian components under $\alpha$.

(…)

References

• Jacob Lurie, Structured Spaces