added pointer to:
]]>Maybe he wanted to make a joke.
As a rule of thumb in mathematical terminology, don’t attribute to elaborate intention what can easily be explained by carelessness. ;-)
Seriously, I’d urge to stick to calling it the (yoga of) six operations. Why change a standard and time-honored terminology without need or improvement?
On the other hand, I doubt that the stray terminology goes back to Gallauer 2021, as the Wikipedia entry currently suggests. For instance our entry six operations lists
Pavel Etingof, Formalism of six functors on all (coherent) D-modules (pdf)
While this (short) file is not dated, it existed already at least in Feb 2014, when I added the link (here).
]]>For anyone wondering about “6-functor”: https://en.wikipedia.org/wiki/Six_operations. And Scholze uses the same terminology, though in the guise of “six-functor formalism”: https://people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf
]]>That’s a good observation :) Maybe he wanted to make a joke.
]]>Got to say that “6-Functor” in the title of Lucas Mann’s paper was somewhat misleading to me; I was wondering why anybody would need to use 6-category-structure preserving functions between two 6-categories in complex analytic geometry.
]]>Added mention of the work of Clausen and Scholze reproving results in complex analysis using condensed mathematics.
]]>Added references to applications.
]]>Added applications subsection.
]]>I only now see that the entry has the following line (dating from 30 May 2022, rev 17, due to the prolific Anonymous):
…current expositions of condensed mathematics rely…
Probably what is meant is not “current expositions” but something like: “existing constructions”. Maybe somebody could change this.
]]>I have touched wording and hyperlinking. Please check if you agree (here).
]]>Added a paragraph to “idea”, explaining how topological abelian groups do not form an abelian category and how condensed abelian groups provide a solution to this problem.
]]>adding sentence about the use of the axiom of choice in condensed mathematics
Anonymous
]]>Yes, in particular, condensed anima are simplicial condensed sets.
]]>For a tad more see at infinity-groupoid: here.
]]>Yes.
]]>In that text linked by David Corfield, are the objects which Peter Scholze calls “-categories” -categories, and the objects which he calls “anima” -groupoids?
]]>Added
Thanks for your candid thoughts, Richard.
]]>Re #9: thanks for the question! I’ll try to avoid de-railing the discussion too much, and just give a brief reply. I don’t wish to stir up any argument, but I will say that I am not all that sympathetic with the main aspects of Tim Campion’s comments. It is great to come up with alternative formulations of things (in terms of Grothendieck fibrations in this case), but I think it is stretching a point to insist that Mochizuki should have used such and such a formulation. There are many reasons to choose a particular formulation of something, not all of which can be easily explained even in ordinary circumstances. In this very specific case, I confess to not being all that fond of wrapping things up into Grothendieck fibrations, so I actually have some sympathy anyhow for avoiding them, but this is really beside the point; I would have made the same argument even if I myself would have chosen to use Grothendieck fibrations.
I do certainly agree that much of the exposition in IUTT may in fact do more harm than good for the moment; but in the long run, if people eventually can understand Mochizuki’s work, this exposition may prove interesting and in hindsight insightful. What is needed for the moment is certainly a more direct and stripped down exposition, but if Mochizuki is not able to provide this, or to understand that his expositionary efforts are for the moment not really helping, that is not a reason to lynch him.
A final point which I have made before, so is not directly a remark on Tim Campion’s comments (although there is something of what I am protesting against there too) is that I think people are not thinking for themselves enough. It is very convenient for people to use Scholze and Stix’s arguments as some kind of ’definitive’ put down of Mochizuki’s work. But one thing is clear to me: Mochizuki’s work without doubt uses anabelian geometry in a fundamental way, in what he refers to as ’algorithms’. This aspect is completely missing in Scholze and Stix’s arguments. If definitive objections are to be made against Mochizuki’s work, the role of anabelian geometry first needs to be understood and explained. In this respect, I think that Absolute Anabelian Geometry III deserves a close reading; there is some very interesting stuff in there, even some homotopies (and viewing of natural transformations as homotopies, which is of course not just a metaphor but is perfectly justifiable)!
]]>It was taken from here.
]]>What is the extra structure related to limit points that Scholze is alluding to?
]]>In discussions on condensed mathematics, I’d been hoping category theorists would have something to say on the set-up. I pointed out here that there’s an apparent similarity with Lawvere’s bornology. Both parties claim to be turning functional analysis into algebraic geometry.
Scholze provided a comparison:
If you wish, condensed sets are bornological sets equipped with some extra structure related to “limit points”, where limits are understood in terms of ultrafilters
I was hoping we’d be able to see whether and how ultrafilters appear naturally here, but nothing came of some prompting.
]]>