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…

]]>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, $i^{!}$?

]]>I think $i_*$ is a right adjoint, so corrected the page at Cahiers+topos#RelationToSyntheticTangentSpaces. Hope my correction is not incorrect!

]]>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.

]]>added pointer to

- Vincent Schlegel,
*Gluing Manifolds in the Cahiers Topos*(arXiv:1503.07408)

both to *manifold with boundary* as well as to *Cahiers topos*

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.

]]>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.

]]>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.

]]>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”

No good reason, I think. I’ll change it. Thanks.

]]>Is there a good reason that ThCartSp redirects to CartSp, rather than to Cahiers topos where it is actually defined?

]]>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 $CartSp_{synthdiff}$.

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?

]]>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?

]]>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.

]]>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?

]]>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.

]]>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?

]]>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.

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.)

]]>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.

]]>Welcome to the n-forum, Eduardo!

]]>&quot;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?&quot;

Well, no. The cahiers topos was the first well adapted model of SDG (a concept that I introduced). I got upset about the topos G whose site of definition are the C^oo rings presented by a germ determined ideal (also a concept I introduced, and that it is the key concept in the theory of well adapted models). This topos is the definitive well adapted model, and actually I introduce it and proved all its basic important properties. I am fond of it. It is the analoge of the Zariski topos of algebraic geometry. Both topoi are defined by the topology of all open covers, which is an essential fact for their good properties. To think that the essential feature of the Zariski topos is that its covers are finite is not understanding what it is going on. Its essential feature is that its covers are all open covers. eduardo dubuc ]]>

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. :-)

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