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created Cahiers topos.
Do I understand correctly that this gadget is named after the journal that Dubuc’s original article appeared in? What a strange idea.
I like it: people who read Cahiers often (like Igor) often refer to notion from there by saying cahiers version etc (I have not heard for this one so far though in these words).
Hm, I don’t know. I would like to use this term now for the oo-version, such as to increase the chance that people know what I am talking about. Should I call it the “(oo,1)-Cahiers topos”??
By the way: a while ago I said that I have a proposal for formalizing when one topos or (oo,1)-topos qualifies as an infinitesimal thickening of another. I said this is the case if the geometric morphism is induced from a morphism of sites that is a coreflective embedding.
Now, for whatever it’s worth, Kock’s notion that he calls “semidirect products of category” $G \ltimes W$ (section 4) is such that the canonical inclusion $G \to G \ltimes W$ is always coreflective, and indeed for him $W$ is the infinitesimal thickening.
Not that this is very deep. But maybe it is good.
This coreflectiveness is exactly what i was pointing in Rosenberg’s work in the parallel discussion, unless I misunderstood the setting. I do not see why you care about bimodules etc. and not the general nonsense part at the beginning.
Okay, so I missed it. On which page do they talk about coreflexivity?
Well, I wrote in the answer there to you that the notion of subscheme (viewed as a category of sheaves) includes the coreflexivity, and that they define a thickenning as the smallest subscheme containing certain things. So it is burried in the sequence of the definitions which resemble the classical setup but written not in terms of rings and spectra but of categroies of qcoh sheaves.
Is the Cahier topos the same model for SDG that Dubuc got upset about at the categories list, that it wasn’t named after him? If so, why isn’t it called the Dubuc topos?
The parallel discussion we allued to is the nForum entry infinitesimally thickened topos. Urs now created nForum discussion on the entry differential bimodule which I started a while ago but never finished.
Is the Cahier topos the same model for SDG that Dubuc got upset about at the categories list, that it wasn’t named after him? If so, why isn’t it called the Dubuc topos?
So the sites for the toposes in question are not equal, but I’d need to concentrate to determine to which extent the toposes are equivalent.
My impression was that in that message Dubuc was remarking that he also invented the toposes that play the main role iin Moerdijk-Reyes’ book (sheaves on finitely generated $C^\infty$-rings), whereas the “Cahiers topos” is about sheaves on infintesimally thickened ordinary things, so about certain very well behaved locally presentable $C^\infty$-rings
I have to say, though, I have not made much progress through the end of the old Sur les modeles. Yet.
What confuses me a bit is that Dubuc in his Sur les models at the end, culminating in theorem 4.10, considers the topos of sheaves on spaces of the form “variété $\times$ infinitesimal” instead of “CartesianSpace $\times$ infinitesimal” as everybody who cites this says.
On page 3 Dubuc mentions variété différentielle for the first time, and seems to take their defnition for granted. I hope he means affine $C^\infty$-variety, then I understand what’s going on. Otherwise there must be fine-print that I am missing,
I hope he means affine C ∞-variety,
I think you’re pretty safe in assuming that. I vaguely recall (and francophones please help me out) that variété is not quite the same as variety in the modern sense. In the older literature (like Poincare) I think this term was meant to describe what we would now call (closed) submanifolds of $\mathbb{R}^n$, i.e. affinely embedded.
Thanks, David, that must be it.
Variété in French may mean a manifold or to mean a variety. Two things in English, one in French.
Variété différentielle means a smooth manifold, not a smooth variety. A smooth variety would have other smooth word: lisse.
Thanks, Zoran. I was going to clarify a little regarding ’différentielle’ - it is obviously a smooth manifold in hindsight.
Okay, but then it all makes sense, too: because the Grothendieck topology on the site with objects of the form $Manifold \times InfinitesimalSpace$ that Dubuc talks about is that where covering families are of the form $\{ U_i \times \ell W \stackrel{p_i \times Id}{\to} U \times \ell W \}$ for $\{ U_i \stackrel{p_i }{\to} U \}$ an ordinary open cover of the maniold $U$ and for $\ell W$ an infinitesmal space.
Since sheaves on the ordinary site of all manifolds are equivalent to sheaves on just Cartesian spaces, this means that also sheaves on the site $\{ Manifold \times Infinitesimal \}$ that Dubuc considers is equivalent to sheaves on the site $\{ VectorSpace \times Infinitesimal\}$ that A. Kock considers, which is in turn equivalent to sheaves on the site $\{ CartesianSpace\times Infinitesimal\}$ that Nishimura considers.
So if Dubuc’s variété différentielle indeed means smooth manifold, then it is clear why all three of these sheaf categories are indeed the “Cahiers topos”.
Okay, thanks. :-)
Welcome to the n-forum, Eduardo!
There are plenty of things in the $n$Lab entries on SDG (see synthetic differential geometry - contents) that deserve to be included, expanded and clarified. This is certainly one of them. Maybe somebody finds the time to add some more. I should do something about it, but I am a bit absorbed with other things at the moment.
A while back somebody had kindly added to Cahiers topos a commented reference to the correction by Kock and Reyes of the article by Kock mentioned there. I have now worked that into the entry more comprehensively and further expanded here and there.
(Notice: the definition in the entry has been the correct definition of Kock-Reyes all along.)
I have added to Cahiers topos a new section Synthetic tangent spaces. So far this just states a basic fact about what the synthetic tangent bundle of a smooth space is, as seen in its canonical reduced embedding into the Cahiers topos.
An object X of the Cahiers topos can be regarded as a smooth prespace whose set of ways of laying out ${\mathbb{R}}^n$ in $X$ is taken as the set of morphisms in the Cahiers topos from (the object of the Cahier topos represented by) ${\mathbb{R}}^n$ to $X$.
For ${\mathcal{C}}$ a differentially good cover of a carteisan space ${\mathbb{R}}^n$, recall the notion of ${\mathrm{GluedPlots}}({\mathcal{C}},X)$, which in any case is a set. Is the canonical set function from $X({\mathbb{R}}^n)$ to ${\mathrm{GluedPlots}}({\mathcal{C}},X)$ given by sending a plot $p$ of ${\mathbb{R}}^n$ to its restriction to all members of the cover ${\mathcal{C}}$ an epimorphism?
The objects in the topos are by definition the sheaves, and the sheaf condition says that the map which you are asking about is an isomorphism, hence in particular an epimorphism.
In definition 2 of covering families of the Cahiers topos, it is required that each element of a cover has the form $p_i\times Id : U_i\times \ell W \to U\times \ell W$. What I’m wondering is if we look at covers whose elements have source $U_i$ a cartesian space instead of $U\times\ell W$, the product of a cartesian space by an infinitesimally thickened point. In this case, do we still get an isomorphism?
A sheaf on whatever site is defined to be a presheaf such that for any space and any cover of that, the map that restricts sections over the space to matching sections over the cover is an isomorphism.
Above you start your question by mentioning “smooth spaces” in the sense of sheaves over the site of just Cartesian spaces, and then later you turn to the site of the Cahier topos, whose objects are infinitesimally thinckened Cartesian spaces. If you look at a sheaf on the latter site, then by definition of sheaf all these restriction maps are isomorphisms.
But from this it also follows that probably it is not clear to me what you are really after. Could you maybe say again in one go what the context is you start with and what in there the question is? sorry.
Here I’m really drawing the analogy with simplicial sets, of which being Kan is a property, and of which being (the nerve) of an ordinary category is a further property. Explicitly, a simplicial set is a presheaf over the simplex category. A simiplicial set is Kan if each horn has a filler. A Kan simplicial set is an ordinary category if each horn has a unique filler.
I try to formulate this in terms of good covers. What does Kan mean? By a cover of a presheaf $X$, mean a family of plots from probes (representable presheaves) to $X$. What does it mean by each horn has a filler? Fix a horn, say the unique inner horn $\Lambda$ of the 2-simplex $\Delta^2$. Fix a cover ${\mathcal{C}}$ of $\Lambda$, say by two 1-simplexes. Consider the canonical map from $X(\Delta^2)$ to ${\mathrm{GluedPlots}}({\mathcal{C}},X)$. To say that $X$ is Kan, is to say that this map is epi. To say that $X$ is a category, is to say that this map is iso.
It seems to me that the enlargements of smooth spaces to objects of the Cahiers topos (and to smooth prespaces) is analogous to the enlargement from (nerves of) ordinary categories to Kan simplicial sets (and to general simplicial sets.)
The feature of Kan simplicial sets is a combinatorial (internal) definition of homotopy groups. Analogously, it would seem that the presheaves of whose canonical map I’m alluding to is epi would allow a geometric definition of homotopy groups in the 1-toposial setting. Is there already a geometric definition of homotopy groups of objects of the Cahiers topos?
Urs, here is a way to ask my question more explicitly. Consider the inclusions ${\mathrm{SmoothSpace}}\hookrightarrow {\mathrm{CahiersTopos}}\hookrightarrow {\mathrm{SmoothPreSpace}}$. If my understanding each correct, then each of this inclusions is actually fully faithful. That is to say, a homomorphism of smooth spaces is just a homomorphism of the underlying smooth prespaces. And a morphism in the Cahiers topos is a homomorphism of the underlying smooth prespaces.
Is there a property of smooth prespaces which makes them objects of the Cahiers topos? For example, a smooth prespace is a smooth space if and only if the canonical map $X({\mathbb{R}}^n)$ to ${\mathrm{GluedPlots}}({\mathcal{C}},X)$ is iso, for any differntially good cover of ${\mathbb{R}}^n$. It is true that an object of the Cahiers topos is a smooth prespace such that this canonical map is epi?
There is some misunderstanding. The way I read the terminology there is no faithful inclusion of the Cahier topos into pre-smooth spaces.
Let’s spell this out, to get to the bottom of where we are talking past each other:
write $CartSp$ for the site of Cartesian spaces, and $CartSp_{synthdiff}$ for the site of formal Cartesian spaces. Then in the terminology that I have used elsewhere
a pre-smooth space is a pre-sheaf on $CartSp$;
a smooth space is a sheaf on $CartSp$;
an object in the Cahiers topos is a sheaf on $CartSp_{synthdiff}$.
Is there a good reason that ThCartSp redirects to CartSp, rather than to Cahiers topos where it is actually defined?
No good reason, I think. I’ll change it. Thanks.
I have split it off as a small entry in itself. Also renamed it to FormalCartSp, in line with FormalSmooth∞Grpd, because, I suppose, that fits traditional terminology better than speaking of “thickening”
Thanks. Maybe “formal” is traditional, but I heartily dislike it; there’s nothing in the English word “formal” that to me suggests an infinitesimal thickening.
Based on the words, I would expect a “formal cartesian space” to be some sort of an abstraction of the ordinary notion of cartesian space.
That’s the common problem with “formal”, true. “formal space” means two completely different things to two different communities. Unfortunately.
In general, the word “formal” in the sense of infinitesimal geometry is a really unwise choice of terminology. But its absolutely standard and widely understood in the respective circles.
added pointer to
both to manifold with boundary as well as to Cahiers topos
34 I do not think that it is “completely different”. Formal is about formal completions, formal power series. They are formal: one can not evaluate the formal sums as functions, they do not converge pointwise, they are often described syntactically as “formal sums of words in symbols” (commutative or not, depending which kind of formal power series we discuss) without evaluation semantics; their grammar is pretty free (in comparison to the grammar of converging series, or functions…). The fact that some formal completion dualize to certain infinitesimal neighborhoods, does not mean that anybody by formal means a synonym for infinitesimal. There are many kinds of infinitesimality, only some are modelled by duals of formal objects.
I think $i_*$ is a right adjoint, so corrected the page at Cahiers+topos#RelationToSyntheticTangentSpaces. Hope my correction is not incorrect!
So then the diagram above your change should have arrows the other way, as we normally have left adjoints on top.
Later on the page, in the proof of Prop 3.9, it speaks of
By general properties of left adjoints of functors of presheaves, $i_{\ast} X$ …
Is this relying on there being a further right adjoint, $i^{!}$?
Thanks. I should have read the rest of the section. It looks like $i_\ast$ was supposed to be $i_!$, the left adjoint of $i_\ast$, throughout that section, according to the usual notation. So I have fixed it (reverted my first edit, and then replaced $i_\ast$ with $i_!$ throughout). Hopefully this is correct…
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