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

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

under construction

**structures in a cohesive (∞,1)-topos**

**infinitesimal cohesion?**

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.

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 $\alpha$.

(…)

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