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David Corfield (Mar 31 2024 at 07:35):Has anyone tried to look to do some sort of 2-algebraic geometry with some 2-category of polynomial functors?
John Baez (Mar 31 2024 at 17:33):In a sense that's what I've been doing with @Todd Trimble and @Joe Moeller, but instead of polynomial functors we're using species, esp. Vect-valued species... and we're focused on the tiny fraction (yet still enormous) that shows up when you look at "polynomials in one variable".
John Baez (Mar 31 2024 at 17:35):So this is like doing algebraic geometry where all you study is the line. It sounds a bit pathetic when you put it that way, but when you do 2-algebraic geometry the line becomes much more interesting: you get the theory of Schur functors, plethysm, symmetric functions, the splitting principle and so on.
John Baez (Mar 31 2024 at 17:39):I do want to go further and connect this up with what James Dolan and I are doing with 2-algebraic geometry using much more general 2-rigs. (The category of Vect-valued species is the free 2-rig on one generator.) James is thinking about modular forms, Artin reciprocity and a lot of other stuff from this viewpoint. But he never writes up anything; instead our progress is visible mainly as YouTube videos, and I haven't even been uploading those for the last half year or so.
John Baez (Mar 31 2024 at 17:43):But maybe none of this addresses your real point, if what you really want is to use actual polynomial functors instead of Joyal's analytic functors (which correspond to species). They're morally similar, but the technical details are so different that they would go in pretty different directions.
Todd Trimble (Mar 31 2024 at 17:54):John Baez said:
In a sense that's what I've been doing with Todd Trimble and Joe Moeller, but instead of polynomial functors we're using species, esp. Vect-valued species... and we're focused on the tiny fraction (yet still enormous) that shows up when you look at "polynomials in one variable".
I feel like I've just walked into a conversation where I didn't quite catch the context, but we do need sometimes to consider species in several variables as well.
John Baez (Mar 31 2024 at 17:58):I think David provided all the context we have in his first comment here!
It's true we do need to look at several-variable species now and then. But only in our new paper in progress do we start heading in the direction of a general study of 2-rigs, which could be called "affine 2-algebraic geometry"... and we're not really tackling it from the viewpoint of 2-algebraic geometry (which is fine).
John Baez (Mar 31 2024 at 18:14):What we're doing is the categorified analogue of studying and a few other 'stand-out' commutative rings, as a precursor to full-fledged affine algebraic geometry... but this categorified analog is so rich, connecting so many subjects, that it's really worth intensive study.
David Corfield (Mar 31 2024 at 19:15):It was just some idle wondering. I was browsing through a talk by Minhyong Kim,
Screenshot-2024-03-31-19.25.19.png
Plenty to think about with your work already.
John Baez (Mar 31 2024 at 19:26):In our forthcoming work we consider some interesting 2-rigs that have a single generator. These are like categorified versions of for some ideal . But many of them are famous things!
John Baez (Mar 31 2024 at 19:27):It would definitely be interesting to explore what happens using Spivak-style polynomial functors, but I don't see them connecting so well to traditional concerns in algebra, topology and geometry. To me they have more of a computer science vibe.
David Corfield (Mar 31 2024 at 20:25):If Minhyong can wonder about arithmetic schemes applying to physics, who knows?
Screenshot-2024-03-31-21.22.57.png
John Baez (Mar 31 2024 at 20:40):Anyway, I'd actually love to talk quite concretely and specifically about 2-algebraic geometry, but I seem have gotten out of the habit of talking about it publicly on the n-Cafe or here, so I have a kind of stewing build-up of thoughts, and all I blog about are my attempts to understand topics in good old 1-algebraic geometry, like motives and elliptic curves and abelian varieties and Grothendieck-Galois-Brauer theory.
John Baez (Mar 31 2024 at 20:42):But Jim's big realization, that affine 2-algebraic geometry subsumes projective 1-algebraic geometry, is at the back of a lot of this.
Brendan Murphy (Apr 01 2024 at 00:46):Does anyone ever compare stuff between affine 2-algebraic geometry and tensor triangulated geometry? I think a lot about the balmer spectrum but I don't think I've heard much about the former, and I'd like to know what lessons from 2AG I'm missing out on. I guess TTG is sort of a higher version, and you have all these nice reconstruction and stratification results there. I know there's reconstruction stuff (Tannaka type things) but I don't know about stratification
David Corfield (Apr 01 2024 at 07:39):In Martin Brandenburg's thesis the terms appear in the same paragraph at least:
Screenshot-2024-04-01-08.29.22.png
I wonder if anyone's developing the approach via the prime spectrum of a symmetric monoidal stable (∞,1)-category rather than via its homotopy category shadow, the Balmer spectrum.
John Baez (Apr 02 2024 at 18:17):I have no idea what tensor triangulated geometry, though I've heard of it. The basic idea of affine 2-algebraic geometry is that it's really good to categorify the concept of (commutative) ring and think about various kinds of (symmetric) 2-rigs - most notably
and more generally the enriched version thereof.
This then subsumes projective algebraic geometry and more, since the category of quasicoherent sheaves on a scheme is a nice example of an enriched 2-rig.
John Baez (Apr 02 2024 at 18:17):If you're interested in this stuff, Martin Brandenburg's thesis is good to read.
Brendan Murphy (Apr 02 2024 at 18:42):Yeah, I definitely should check it out
Brendan Murphy (Apr 02 2024 at 18:48):Tensor triangular geometry is the same idea, but instead of representing a scheme X by its category of quasicoherent O_X-modules (which is symmetric monoidal abelian w/ a tensor product preserving arbitrary colimits) we replace it by the derived category of complexes of O_X-modules with quasicoherent cohomology (which is symmetric monoidal triangulated with tensor product preserving exact triangles and arbitrary direct sums, or is a symmetric monoidal stable infinity category with tensor product preserving arbitrary colimits). There's a small and a large version of this which is roughly about whether you study the compact objects or all the objects (we can go between the two via ind completion). A small tensor triangulated category has a spectrum defined in terms of prime ideals like the zariski spectrum of a ring, and for a qcqs (eg noetherian) scheme the spectrum of its derived category of perfect complexes recovers the scheme
Brendan Murphy (Apr 02 2024 at 18:49):Although I guess this is only the enriched version of what you were talking about (enriched over the stable homotopy category, or for things arising from AG enriched over D(Z)?)
John Baez (Apr 02 2024 at 23:17):Thanks! I find triangulated categories scarily technical so I avoid them, but working with the derived category of complexes of O_X modules seems reasonable to me on general grounds, as a kind of approximation to working with "-rigs" (which also seem scarily technical to me - or would if anyone has defined them).
Brendan Murphy (Apr 03 2024 at 00:24):Yeah, triangulated categories suck and although enhancements for them are a lot less sucky they're still technical
Kevin Carlson (aka Arlin) (Apr 03 2024 at 17:31):I’m pretty sure the spectrum of a stable -category is the same as the spectrum of its underlying triangulated category, by the way. It’s kind of cool, actually, that you take certain “prime ideals” in a suspiciously literal sense, using certain subcategories and tensor instead of subsets and the ring product, and yet you get something that’s actually rather useful. This is what my advisor was actually working on while I ran off and did my own almost totally unrelated thesis.
Brendan Murphy (Apr 03 2024 at 18:56):Hahaha I was actually really surprised when I looked at your thesis a little while ago and saw your advisor was Balmer