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Basic instructions before leaving Earth?

5 December, 2020

It should be fairly clear that I don’t blog here as much as I once intended, and that even when I do there isn’t really enough mathematics. This post is intended as a placeholder or index for any future posts that I might write which relate to the theory of “liquid vector spaces” that emerge from Clausen & Scholze’s framework of “condensed mathematics”.

My interest was sparked by

  • (n-category cafe, 2020-03-28) Corfield: Pyknoticity versus cohesiveness (see discussions in comments)
  • (Xena project, 2020-12-05) guest post by Scholze: Liquid tensor experiment (which is very lucid and even-handed in its exposition, as well as being interesting in the context of Buzzard’s larger project)

since for a few years during and after my PhD studies, I became interested in what might be the “proper” setting for the homological approach to Banach bimodules over Banach algebras as pioneered by Helemskii and his Moscow school. This interest led me to observe, over the years, a number of people independently realise or propose that Functional Analysis as classically defined is working in “the wrong category”, and to then claim with varying degrees of tongue-in-cheek badinage or missionary zeal that Y’all Only Have Difficulties Because Y’all Insist On Doing Things The Wrong Way.

My perspective at the time is alluded to in this 2007 post. Despite the mild snark above, I am actually by the standards of functional analysts quite partial to “abstract nonsense”, as one might detect from my first published paper; but at heart I still get a kick from inequalities and explicit formulas, and remain a believer that it is worth knowing how to take a long but low-tech approach to given problems.

Tangentially, I still want to see a proof of the known result that the “tensor-flat” Banach spaces are precisely the “script L-1” spaces of Lindenstrauss and Pelczynski, which somehow runs parallel to the proof of the Govorov-Lazard theorem that flat modules are filtered colimits of finitely generated free modules. (There was something along these lines in work of Aristov, IIRC, but I never quite satisfied myself that it was what I was after.) Perhaps this would follow from a suitable embedding of the category of Banach spaces and “short” linear maps into the category of liquid vector spaces and whatever-the-morphisms-are?

Minor update 2020/12/13: upon re-reading Scholze’s post, I couldn’t help doing a double-take when I saw this passing remark:

When combining bounds with homological algebra, it means that within all our complexes we have to carefully keep track of norms. This quickly gets nasty. Did you ever try to chase norms in the snake lemma? In a spectral sequence?

Erm … yes? I mean, perhaps not in such a fancy setting, but see e.g.

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