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At synthetic differential infinity-groupoid I have entered statement and detailed proof that flat and infinitesimally flat real coefficients are equivalent in $SynthDiff\infty Grpd$
$\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \simeq \mathbf{\flat} \mathbf{B}^n \mathbb{R} \,.$The proof proceeds by presentation of $\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}$ by essentially (a cofibrant resolution of) Anders Kocks’ s infinitesimal singular simplicial complex. In this presentation cohomology with coefficients in this object is manifestly computed as in de Rham space/Grothendieck descent-technology for oo-stacks.
But we also have an intrinsic notion of de Rham cohomology in cohesive $\infty$-toposes, and the above implies that in degree $n \geq 2$ this coincides with the de Rham space presentation as well as the intrinsic real cohomoloy.
All in all, this proves what Simpson-Teleman called the “de Rham theorem for $\infty$-stacks” in a note that is linked in the above entry. They consider a slightly different site of which I don’t know if it is cohesive, but apart from that their model category theoretic setup is pretty much exactly that which goes into the above proof. They don’t actually give a proof in this unpublished and sketchy note and they work (or at least speak) only in homotopy categories. But it’s all “morally the same”. For some value of “morally”.
Urs, link does not work: Grothendieck descent, rather than Grothenndieck with two n-s.
But it’s all “morally the same”. For some value of “morally”.
It is very often than modern mathematicians use that term, but an excellent teacher of mine with logic training and fine linguistic feeling, was complaining to me that it is a bad choice to attach morality as a question of will, value, consensus or attitude to a variant of truth. Truth could be explained or weighted roughly, vaguely, imprecisely, approximately, essentially, but morally, this would mean depending on the human consensus, not about the essence of things. Since that critique, which I appreciated, I rarely used the term myself. Essentially the same is not subject to the same critique, I believe.
I hope you enjoy his thought.
I always thought “morally the same” in math is meant as in “the moral of the story”: we learn the same lesson.
The are also worse variants like “this is morally correct” for a mathematical proof where the “lesson” interpretation can hardly pass.
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