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A few times here we had mentioned the thought of looking at models of differential cohesion where the “infintiesimal shape” contracts away germs instead of just formal neighbourhoods.
It’s only via her message to the categories-mailing list today that I become aware that this seems to be what Marta Bunge et al. call “synthetic differential topology”. From the introduction to the book of that title available here :
Whereas the basic axioms of SDG are the representability of jets by tiny objects of an algebraic nature, those of SDT are the representability of germs (of smooth mappings) by tiny objects of a logical sort introduced by Jacques Penon
The book is scheduled to appear April 2018. I am getting the impression that there is no related document available?
Presumably what they’re doing in the book is closely related to Synthetic aspects of C∞-mappings II: Mather’s theorem for infinitesimally represented germs.
Thanks! That confirms my impression from #1.
So I gave it a stub entry: synthetic differential topology.
In that table you have germs lined up with the arithmetic ’localized at’, and this already features in arithmetic cohesion. E.g., at torsion approximation
$ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}.$So why do we not have germs at play already with existing cohesive modalities?
[Lots of symbols not displaying at torsion approximation , arithmetic cohesion – table. They needed fixing with ʃ]
So why do we not have germs at play already with existing cohesive modalities?
That’s what I was referring to in #1: I suppose it should be true that the sequence of infinitesimal shape modalities $\Im_{(n)}$ that contracts away order $n \in \mathbb{N} \sqcup \{\infty\}$ infinitesimal neighbourhoods may be continued with $\Im_{germ}$ which “contracts away germs”.
Lots of symbols not displaying
I had alerted Adeel a while back. Maybe he is too busy. Or maybe you write to him and remind him.
Does the approach in that paper in #2 offer any hints? They’re representing germs as mappings on infinitesimal domains, and these are $\Delta^n$ which are tiny, where $\Delta = \neg \neg \{0\}$ in the intrinsic open topology.
I fixed the symbol issue. It was a case of use the right esh symbol.
I am getting the impression that there is no related document available?
An early draft of the book was posted to ResearchGate by Martha Bunge around August of 2016. It looks like it was taken down, but I still have the PDF. I don’t want to post it online, but I wouldn’t mind sending it to you, if you wanted.
I’d be interested to see it. Contact details at David Roberts.
Another paper which gives the idea is Bunge and Dubuc, Local Concepts in Synthetic Differential Geometry and Germ Representability.
If the object representing germs is $\neg \neg \{0\}$, does this mean $\sharp$ as $loc_{\neg \neg}$ is involved?
Why in dcct (def 4.2.7) does $\Im$ appear as $\Im_{(0)}$? Infinitesimal neighborhoods grow with the order of the infinitesimals, up to $D_{\infty}$ for all nilpotent elements. Isn’t it localizing at paths in the latter that $\Im$ amounts to?
Here (#5) we’re wondering about taking things further by contracting away germs, so $\Im_{germ}$ is beyond $\Im_{(\infty)}$. So shouldn’t dcct (def 4.2.7 and following diagram) have it the other way around?
Maybe. Thanks for the alert. I’ll check later.
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