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I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.
Some additions to regular differential operator in noncommutative geometry.
I corrected the plural redirect topologizing subcategories. Previous calls were failing as the redirect line in the entry read “toologizing subcategories”. It would be interesting to see the comparison between the picture here and in the setup of Cahiers topos. I mean, various conditions, on the subcategories corresponding to infinitesimal thickenings.
Here is also archived an old dicussion from infinitesimal object:
Do all of the following really involve infinitesimal objects? Or should we move the others to infinitesimal quantity to clarify that this page is about objects of categories as defined below? —Toby
Urs Schreiber: good point. I think Lawvere’s definition is, as stated, to be thought of as defining _infinitesimal space_s, yes. For infinitesimal quantities it will likely have to be dualized (in the hopefully obvious way).
So maybe we should rename the entry here into infinitesimal space and, yes, create another entry on infinitesimal quantities. Yes, I think that’s a good idea. I have to run now, but I can implement it later.
Toby: I could do it too, although I'd like to see what Zoran thinks.
Zoran: I don’t really feel/know what is the most sensitive here. Surely our understanding is developing and we will see more in future (you see yet another thing are the sheaves supported on infinitesimal neighborhoods as well as the duality between infinitesimals and (regular) differential operators in algebraic setting, the duality whose analogue I do not understand in the smooth context (cf. Maszczyk 0611806).
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