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


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

under construction



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

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

A:(𝒪 𝒳R):𝒳R𝒪 𝒳H 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 𝕋jH𝒪 𝒳𝒳 \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\mathcal{C}_{\mathbb{T}} of 𝕋\mathbb{T} with the inverse image 𝒪 𝒳\mathcal{O}_{\mathcal{X}} as exhibiting a structure sheaf of local 𝕋\mathbb{T}-∞-algebras in 𝒳\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 𝕋\mathbb{T}-algebra-ed (,1)(\infty,1)-toposes. This extra datum is encoded by a choice of morphisms in H\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 H\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.


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\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 𝒳\mathcal{X} a T-structure sheaf on 𝒳\mathcal{X} is a geometric morphism

A:𝒳H A : \mathcal{X} \to \mathbf{H}

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

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

𝒳 𝒪 𝒳 f * α H 𝒪 𝒴 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:YXp : Y \to X in H\mathbf{H} all its component naturality squares

f *𝒪 𝒳(Y) α Y 𝒪 𝒴 f *𝒪 𝒳(X) α X 𝒪 𝒴 \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 YXY \to X is of the same, but converse form: given an infinitesimal cohesive neighbourhood

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

we have canonically given a natural transformation

ϕ:i !i * \phi : i_! \Rightarrow i_*

looking like

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

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

i !X ϕ Y i !Y i *X ϕ Y i *Y \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

𝒳 𝒪 𝒳 i ! f * α H ϕ H th 𝒪 𝒴 i * 𝒴 \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\mathbf{H} that have cartesian components under ϕ\phi also have cartesian components under α\alpha.



Last revised on March 5, 2012 at 23:42:01. See the history of this page for a list of all contributions to it.