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added some lines to differential algebraic K-theory
also a stub Beilinson regulator
So is $Mfd$ going to be the category of choice when someone wants to refine a cohomology to a differential one? ’Just’ multiply the original site by it, and take $\infty$-stacks (perhaps with different values).
Presumably along with this will come the super- and synthetic refinements. Is there any chance to infinitesimally extend $Sch$ by itself?
Yes, this is a thought is hard not to have here, that maybe cohesion modeled on manifolds and their slight variants is kind of canonical, and everything else is obtained by doing this over different base $\infty$-toposes.
But one should be careful here. Just because this is maybe easiest to understand for “us” at this point in time, need not mean that this is canonical on more absolute ground. Let’s keep an open mind about this.
Concerning the second question: schemes already involve infinitesimal thickening. It is this crucial observation of Grothendieck’s that spectra of nilpotent rings model infinitesimal spaces which Lawvere then meant to abstract away from the algebraic context by the axioms of synthetic differential geometry.
So we might have had some infinitesimal cohesion already of $\infty$-stacks over $Sch$ relative to those over schemes for non-nilpotent rings.
Concerning the second question: schemes already involve infinitesimal thickening. It is this crucial observation of Grothendieck’s that spectra of nilpotent rings model infinitesimal spaces
I think one has to be careful with this formalization here. The datum for an infinitesimal thickening is implied by the data of an embedding of scheme. Say a diagonal in $X\times_S X$. But the diagonal with a thickening is not a scheme any more. It is rather a formal scheme; formal schemes themselves form a subcategory of the category of ind-schemes (ind-objects in the category of schemes). Similarly with formal spectra of complete local rings and more general topological rings (say pseudocompact rings).
In our paper with Nikolai Durov, instead of $CRing^{op}$, he takes as a base “site” the opposite to the category of pairs $(R,I)$ where $R$ is a ring and $I$ is a nilpotent ideal (later he looks at slice category over $(k,0)$ where $k$ is a ground field; in this category he find two different analogues of the ring of dual numbers, hence gets different kind of formal thickenings). In the corresponding category of presheaves he finds formal schemes as examples. He does not go into topologies to single out the subcategory of formal schemes but works with general presheaves and then with examples which are obviously formal schemes. This enables him to go beyond the classical case of very nice base rings for formal scheme theory.
I should have said “formal scheme”, I guess. The point is that Grothendieck realized and/or amplified that it’s nilpotent elements in rings of functions that serve to model infinitesimal extension.
(In a way this was clear ever since Leibniz’s differential caclulus and many experimental physicists do all their differentiation needs this way without ever having heard of Grothendieck, but I guess he was the one to establish the fact in the maths community conciousness.)
I have started to add some more genuine content to differential algebraic K-theory. But the entry remains rather skeletal and under construction for the moment.
added publication data for all these items:
Ulrich Bunke, Georg Tamme, Regulators and cycle maps in higher-dimensional differential algebraic K-theory, Advances in Mathematics Volume 285, 5 November 2015, Pages 1853-1969 (arXiv:1209.6451, doi:10.1016/j.aim.2015.08.004)
Ulrich Bunke, David Gepner, Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory, Memoirs of the American Mathematical Society 2021 (arXiv:1306.0247, ISBN:978-1-4704-4685-7)
Ulrich Bunke, Georg Tamme, Multiplicative differential algebraic K-theory and applications, Ann. K-Theory 1(3): 227-258 (2016) (arXiv:1311.1421, doi:10.2140/akt.2016.1.227)
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