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The axiomatics of Lurie’s Structured Spaces is mostly just the evident $\infty$-category theoretic version of something that could well have – and maybe should have – been axiomatised way back at least in the Elephant: the concept of a “locally algebra-ed topos” as a basis for axiomatic geometry, being a tautological spin-off of the theory of classifying toposes and the way this theory bridges between algebra and geometry.
But in Structured Spaces there is one key additional ingredient: the notion of open maps = “admissible morphism” which in terms of the classifying topos language is the notion of geometric structure (def. 1.4.3): a natural factorization system on $Topos(\mathcal{X}, \mathcal{K})$, natural in $\mathcal{X}$, where $\mathcal{K}$ is the classifying topos for the given notion of local algebra.
Given this, the subcategory of $Topos(\mathcal{X}, \mathcal{K})$ on the right part of the factorization system is what actually is being called the category of $\mathcal{K}$-structure sheaves on $\mathcal{X}$.
I want to see how this connects to cohesion and infinitesimal cohesion. I suspect that there should be a way to speak of that additional information (openness, admissibility, geometric structure) that ought to be there on top of the classifying-topos-yoga in terms of the extra adjunctions provided by cohesiveness.
For consider this:
in the case of the étale site (section 4.3), the admissible morphism are, of course the étale morphisms;
one way to say etale morphism is to say:
But notice: for the first condition in the second point here, Kontsevich-Rosenberg gave an adjunction characterization, which we noticed is a characterization in the general context of infinitesimal cohesion.
So this means that in the case that the classifying topos $\mathcal{K}$ (which might be our preferred cohesive topos) is equipped with the structure of infinitesimal cohesion $i : \mathbb{K} \hookrightarrow \mathbb{K}_{th}$, then we are entitled to say a morphism $f : X \to Y$ in $\mathcal{K}$ is a formally etale morphism if
$i_! X \to i_! Y \prod_{i_{*} Y} i_{*} X$
is an equivalence.
I am not sure if there is an analogous general abstract formulation of local presentability, and if it is even needed. But my hunch now is that we ought to be able to say that
If a classifying topos $\mathcal{K}$ is equipped with the extra structure of infinitesimal cohesion $i : \mathcal{K} \to \mathcal{K}_{th}$ then $Topos(\mathcal{X}, \mathcal{K})$ is canonically equipped with a factorization system and we define $\mathcal{K} LocAlg(\mathcal{X})$ to be the subcategory spanned by the right morphisms in this system.
Or something like this.
To strengthen the above point:
I think it is clear that with respect to the notion of infinitesimal cohesion given by the inclusion
$i :$ SmoothooGrpd $\hookrightarrow$ SynthDiffooGrpd
a morphism of manifolds $f : X \to Y$ is formally étale in the Rosenberg-Kontsevich sense that $i_! X \stackrel{\simeq}{\to} i_! Y \prod_{i_* Y} i_* X$ precisely if it is a local diffeomorphism.
So that’s the right kind of morphism that one wants to identify as “admissible”.
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 $(f, \alpha) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{X}, \mathcal{O}_{\mathcal{X}})$
$\array{ \mathcal{X} \\ \uparrow & \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}} \\ {}^{\mathllap{f^*}}\uparrow &{}^{\mathllap{\alpha}}\neArrow& \mathbf{H} \\ \uparrow & \swarrow_{\mathrlap{\mathcal{O}_{\mathcal{Y}}}} \\ \mathcal{Y} }$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 $\alpha$.
Written this way this looks like it might be telling us something. The question is: what? :-)
I do not know what kind of characterization of locally of finite presentation you seek for; in Kontsevich-Rosenberg NcSpaces 5.12.1 there is a sketch of the classical approach in categorical language (coming eventually from SGA; it is basically the expression of compactness in the slice category setup). There is also a treatment of formally open $\mathbf{A}$-immersions in 5.11. Probably you noticed those already, but just quick response to your request in 1.
P.S. locally of finite presentation is an adjectival phrase so not in accord with $n$Lab conventions. Maybe we want to have morphism of finite presentation which would also cover more basic morphism locally of finite presentation. Removing locally here means adding additional finiteness conditions. The finitely presented object is nothing but a compact object and morphism thing is the appropriate relativization of the notion.
Thanks, Zoran.
In fact I did not notice the part if RK that you mention. I’ll have a look.
I am wondering if I should worry about local finite presentation. Currently I feel like simply requirig formal étalness and be done with it. That seems to have all the nice relevant abstract properties needed.
But maybe I am wrong.
locally of finite presentation is an adjectival phrase so not in accord with nLab conventions. Maybe we want to have morphism of finite presentation
right, good point
Well the descent and cohomology related notions are well defined e.g. for faithfully flat case, no need to have fppf or fpqc. But for any real computation and richer geometry it is almost impossible without working within some finiteness conditions. Probably it is the same for etaleness stripped from the finiteness conditions.
I mean, say you are in smooth spaces. Now you look at the properties of morphisms. For a smooth cover of a smooth manifold you want that the domain be a smooth manifold as well, not some “infinite-dimensional” smooth space. If you know that the domain is a smooth manifold than of course, the formal smoothness will be the same as smoothness. And similarly for other properties. Your base can be also infinite-dimensional, say some moduli space, but you want to look a space over it which is finite in some sense over it. I think that you silently assume the domain when you test the notions against intuition so that makes you wonder about why it does not look that the finiteness conditions have to be applied on a morphism. No ?
One more aspect about infinitesimal thickenings which may also be expressed via cartesianess of similar kind is the completion.
For example in our paper with Durov, in 7.11 in
he looks at the completions along morphisms in certain category $E$ of covariant presheaves on the category whose objects are pairs $(R,I) = (ring, nilpotent ideal)$. (In fact he considers a slice category over the base ring, but this is not essential here, I think). In particular he considers the natural map $\pi_{(R,I)}: (R,I)\to (R/I,0)$. Then he says that a morphism $\phi:H\to F$ is complete if the square
$\array{ H(R,I)&\stackrel{H(\pi_{(R,I)})}\longrightarrow&H(R/I,0)\\ {}^{\mathllap{\phi_{R/I}}}\downarrow&&\downarrow {}^{\mathrlap{\phi_{R/I}}}\\ F(R,I)&\stackrel{F(\pi_{(R,I)})}\longrightarrow&F(R/I,0) }$is Cartesian. Then every morphism $H\to F$ factorizes as $H\stackrel{u}\longrightarrow \hat{F}_H\stackrel{\kappa}\longrightarrow F$ where $\kappa$ is complete and which is universal among all such; namely the component $\kappa_{(R,I)}:\hat{F}_H(R,I)\to F(R,I)$ is obtained as a pullback of $\phi_{R/I}$ along $F(\pi_{(R,I)})$.
This completion is an example of taking the infinitesimal neighborhoods in geometry. (Here presheaves $H$ and $F$ are viewed as generalized spaces, as usual, and $H\to F$ is typically a monomorphism.) Recall that in the usual formalism of formal schemes (which are certain class of ind-objects in schemes), one has a topological space with a structure sheaf of topological rings, whose stalks are in the classical case the complete local rings.
I mean, say you are in smooth spaces. Now you look at the properties of morphisms. For a smooth cover of a smooth manifold you want that the domain be a smooth manifold as well, not some “infinite-dimensional” smooth space. If you know that the domain is a smooth manifold than of course, the formal smoothness will be the same as smoothness. And similarly for other properties. Your base can be also infinite-dimensional, say some moduli space, but you want to look a space over it which is finite in some sense over it. I think that you silently assume the domain when you test the notions against intuition so that makes you wonder about why it does not look that the finiteness conditions have to be applied on a morphism. No ?
Yes, I think that’s right.
I guess the thing is that I am not currently after studying étale morphisms in general, but am just focusing on an intrinsic description on étale morphisms on objects in a given site of definition for the ambient big topos. I want an intrinsic way to identify the “admissible” morphisms in a geometry (for structured (infinity,1)-toposes).
Right, somehow, I am not yet through real understanding of admissibility in known contexts, though few times it looked to me that I got it.
I am not yet through real understanding of admissibility in known contexts,
I believe you should read étale for admissible throughout.
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