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Some of you may remember that a while ago I had started wondering how one could characterize geometric morphisms of toposes $E \to F$ that would exhibit $E$ as an “infinitesimal thickening” of $F$.
Instead of coming to a defnite conclusion on this one, I worked with a concrete example that should be an example of this situation: that of the Gorthendieck toposes on the sites CartSp and ThCartSp of cartesian spaces and infinitesimally thickened cartesian spaces.
But now I went through my proofs for that situation and tried to extract which abstract properties of these sites they actually depend on. Unless I am mixed up, it seems to me now that the essential property is $CartSp$ is a coreflective subcategory of $ThCartSp$ and that in the respective adjunction
$CartSp \stackrel{\leftarrow }{\hookrightarrow} ThCartSp$buth functors preserve covers.
So maybe it makes sense to take this as a definition: a geometric morphism of Grothendieck toposes is an infinitesimal thickening if it comes from such a coreflective embedding of sites.
Details of this, with more comments on the meaning of it all and detailed proofs, I have now typed into my page on path oo-functors in the section Infinitesimal path oo-groupoids.
In 1-categorical abelian situation, Rosenberg looked at a notion of differential endofunctors, and they depend on some choice in general, because for general categories there is no analogue like the closedness of the diagonal. This is all in 2 preprints
He takes a minimal subscheme containing the identity endofunctor in the category of endofunctors. Subschemes in his abelian setup are coreflective topologizing categories. Toplogizing is an appropriate condition to do the localization theory properly in his setup. That is why i respond above to your insight that coreflectiveness in infinity situation is important for infinity Lie theory.
Part I of the Rosenberg's papers listed above is sufficient. Part II is not of your concern I see now.
The general nonsense part starts at page 19 of part I (pdf) esp. 3.1 and uses some of the localization and subscheme definitions between pages 13-19, especially things from 2.7.3 page 18. See also Prop. 4.3.2.
Thanks, I have found it now. Hm, not sure yet. He seems to use co-reflectivity to define the general notion of “sub-space”.
I was thinking of it as modelling concretely the infinitesimal thickening. The point being that if $X$ is an ordinary piece of space and $D$ an infinitesimally thickened point, then every morphism $X \to D$ factors through the point, while there are nontrivial morphisms the other way round $D \to X$, of course.
So this means that morphisms of the form $X \to Y \times D$ are in bijection to morphisms $X \to Y$. So the embedding of ordinary spaces $X$ into those with an infintesimal thickening $D$ is left adjoint to the projection $Y \times D \mapsto Y$ that forgets the infinitesimal part.
This is effectively (albeit in different words and symbolds) also what happens in section 4 of A. Kock’s Cahiers note .
Is anything of this sort in Rosenberg’s work?
Not of subspace. This is not true. Of subscheme. And he says that the infinitesimal thickening oif the diagonal is the smallest subscheme of the abelian category of additive endofunctors containing the identity endofunctor. But then also he considers thickenning of other choices of generalized diagonal.
So usual differential operators come from thickennings of the diagonal (like in theory of crystals) while you can consider other embeddings and thickenings of those.
Look also at 2.7.4 in terms of Gabriel multiplication.
Is anything of this sort in Rosenberg’s work?
Look at proof of lemma 3.1.1.
By the way, what I do not like about the ringed topoi picture with infinitesimals is that it requires the ring structure giving D. It should be completely in terms of categories not generators.
Okay, thanks for the pointers. I’ll try to look again.
By the way, what I do not like about the ringed topoi picture with infinitesimals is that it requires the ring structure giving D. It should be completely in terms of categories not generators.
Yes, absolutely. This is why I was looking for a more intrinsic way to say “infinitesimally thickened topos”.
So the embedding of ordinary spaces $X$ into those with an infintesimal thickening $D$ is left adjoint to the projection $Y \times D \mapsto Y$ that forgets the infinitesimal part.
Recall the meaning of the module of Kahler differential, which for general thick subcategory T corresponds to a generalized conormal bundle. Then in LR 3.2 (page 20) you have a morphism in formula (1) having a similar role.
Look at proof of lemma 3.1.1.
Hm, now that I look at it: he says “co-reflective” when the inclusion functor has a left adjoint. This is what I know as a reflective subcategory. ?! Co-reflective meaning that it has a right adjoint.
This is a typo. See LR 2.1: The subcategory Y is coreflective which means that the inclusion functor Y to X has a right adjoint.
I need to think about this. Direct comparison seems to be hard, because the co-reflexivity that I am thinking about is at the level of sites , whereas Rosenberg talks about abelian categories of sheaves.
But this is more intrinsic: saying everything in terms of the category of sheaves abelian or not.
Yes, that’s true of course. But I don’t yet understand it at that level! ;-)
If I understood correctly the usual differential calculus corresponds to the conormal bundle of the diagonal which is the identity endofunctor, while for other embeddings the infinitesimal thickenings come from applying the machinery to the corresponding abstract conormal bundles in the sense of LR.
But I don’t yet understand it at that level!
I am sure you will understand it soon ! When we met first time in Bonn in 2005 I was teaching you what a fibered category is, and when Igor first time came to Hamburg 2006 he was teaching you what is a Quillen model category and you were complaining why he needs talking topos all the time and now you are miles beyond our understanding of these things. :)
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