the linked article Weil algebra is about Weil algebras in Lie theory rather than Weil algebras in synthetic differential geoemtry, so replaced link with link to infinitesimally thickened point.
Anonymous
]]>with the recent expansion of the chapters on categories and toposes and on smooth sets I am now streamlining the existing discussion here on supergeometry.
For instance, the proof of the super-differential-cohesive system of adjoint functors on SuperFormalSmoothSet (this prop.) used to try to prove it all from scratch. I have rewritten it now, just citing the proof of cohesion and of differential-cohesion from the earlier chapters, and instead expanding out the arguments for the super-aspect more.
]]>Re #45, is there nothing to be gained from Jos’s smooth rings I mentioned in #29?
]]>Regarding the that you point to: this is indeed related, but it’s not the answer to an open question.
Namely for an -ring then of its Picard -groupoid is the group of “shift twists”, equivalently it classifies those -modules which are just shifted copies of . Hence for a -periodic then this is . If we take to be even 2-periodic then this is . corresponding to the fact already appealed to above, that we have the “-lines” and .
]]>Oh, now I see what you mean to say!
Right, but, you see, the problem is not in knowing what a parameterized morphism of -algebras would be, parameterized over an infinity-groupoid. That’s trivial. Here the problem is to say what it would mean to parameterize smoothly over a smooth manifold.
Consider the simple case that the domain ring is the sphere spectrum. Then homomorphisms into any are just elements of . And so a parameterization now is just a family of elements of . But is just an -algebra, there is a priori no smooth structure on it in any sense, so what would be a smooth family of elements? This needs some extra concept to make sense of.
Of course one standard answer is to just tensor the codomain with the algebra of smooth functions on the given parameter space. But if we tensor over an -algebra, then all the torsion in the -algbra disappears, it collapses equivalently to just a dg-algebra, and so the whole point of invoking spectral geometry disappears.
So that’s why making sense of the category of objects “” needs some extra idea.
]]>I was hoping to shed some light on your decompostion:
consisting of 1) a smoothly -parameterized morphism of -algebras over (not sure yet though what “smoothly parameterized” should mean) and 2) a morphism .
and further hoping that Harpaz’s decomposition might help:
Probably not, then.
While I’m on the trail of likely unhelpful literature, was there something in what Sagave is doing in arXiv:1111.6731 on the second page about the bottom homotopy group being , or is that kind of consideration the reason you brought up 2-periodicity in the first place?
]]>Yes, but what’s the relation to what we are discussing? I don’t understand what you mean to point me to.
]]>Sorry, probably late in the day, is that at the end of the second answer not about maps between parameterized -algebras (admittedly over the same )?
]]>Hm, these links lead to discussion of parameterized spectra. What should be the relevance for the present discussion?
]]>Is there anything helpful in this MO answer or this follow-up MO answer?
]]>Yes, but the other way around: fixing an even periodic -ring , then affine super--schemes would be the formal duals of -algebras over . Among these, there are the super-Cartesian -schemes
(where I am falling back to writing for what I previously denoted , the free -symmetric -algebra construction on an -module).
Now I am thinking: by direct analogy to what we said yesterday in the other thread regarding extensions of in the category over , I suppose it should follow that also the extensions of (say in formal duals of Hopf -algebras over ) would be just the brane bouquet as over , but with all coefficients of replaced by (and all degree shifts replaced by ). Maybe I am missing something here, but it seems to me this conclusion must follow just from the defining free property of those .
So maybe more interesting would be to look for an “ superpoint” given by an -algebra (over ) that looks a little like , but is not just built freely this way. Maybe there is something like this that arises naturally somewhere.
Apart from this what needs thinking is how to merge the -geometry with the cohesive geometry. I vaguely have the thought that we should consider as site the -category whose objects are formal products
“”
of a Cartesian space with an affine super--scheme (the latter in place of an infinitesimally thickened point over as in the construction of the Cahier topos), and whose morphisms are pairs, consisting of 1) a smoothly -parameterized morphism of -algebras over (not sure yet though what “smoothly parameterized” should mean) and 2) a morphism .
The second component, due to the tensoring over , will collapse the -algebra equivalently to a super-dg-algebra over , but the first part would retain genuine -ring theoretic information.
]]>And we might then form and then super--spaces?
]]>Here is the suspension operation on underlying spectra. On the level of homotpy groups this is simply the degree shift by +1. Hence this makes be the analog in -algebra over of for ordinary algebra over .
Then forms the free commutative -algebra over this module (I was following the notation in Charles’ arXiv:0902.2499, p. 10 and later). This is the homotopy analog of saying that the Grassmann algebra is the free graded-symmetric -algebra generated from .
In any case, superpoint or not, I suppose the answer to my puzzlement above is: the homotopy-theoretic analog of superalgebras is not general -rings, but -algebras over even 2-periodic -rings.
I have added a comment to this extent to geometry of physics – superalgebra, starting here.
]]>I’ll need to have this unpacked this a bit. Why the ’’? What does do more explicitly?
]]>So how about -superpoints? Let be an even 2-periodic -ring spectrum. For an -module spectrum, write for the free -algebra over that it generates.
Then is like an -version of the Grassmann algebra over on generators, and
would be the corresponding -theoretic super-point.
I suppose.
]]>Something related to this is the “-graded formalism” of Charles Rezk’s arXiv:0902.2499 (section 2), for -algebras over even periodic -rings.
]]>A possible solution to the issue that -geometry is by default -graded in degree 0, instead of -graded, might be to observe that -graded geometry sits in -graded geometry by the construction that regards a -graded commutative algebra as a periodically -graded algebra (this embedding is faithful, but not full).
Hence we should maybe be looking for a “brane bouquet” starting with for a periodic ring spectrum.
]]>Presumably one needs some kind of K3 technology (eg K3 cohomology), since a K3 surface minus 24 discs is the relation in framed cobordism expressing the ’mod 24’.
]]>In view of
Kapranov’s arguments for his suggestion (the super-representation categories and the appearance of for the string)
and the difficulties untangling the different instances of , maybe it would make sense to see what the has to do with the superpoint.
You say at super algebra that one route is via M2 branes ending on a M9 brane, but the latter doesn’t show up on the brane bouquet.
]]>Has there been any uptake of Christopher Schommer-Pries and Nathaniel Stapleton’s ’superalgebraic cartesian sets’?
Cohesion is in the air (p.11) Rational cohomology from supersymmetric field theories.
Since the category has all finite products the category of superalgebraic cartesian sets is a cohesive topos
So then the site is infinity-cohesive:
]]>Example 3.1. The site for a presheaf topos, hence with trivial topology, is -cohesive, def. 2.1, if it has finite products.
From the ring of functions side, how does one get led to manifolds and supermanifolds? There’s the idea of smooth algebras or -rings, and then smooth superalgebras or -superalgebras.
What happens in the case? I see Jos Nuiten (thesis) has smooth as rings with respect to the smash product on spectra in . Are there smooth super-, replacing by ?
Is there a Fermat theory approach? We have super Fermat theories in arXiv:1211.6134:
In light of the history of -rings and their role in synthetic differential geometry, it is natural to believe that super Fermat theories should play a pivotal role in synthetic supergeometry, but we do not pursue this in this paper.
So there should be a
super Fermat (∞,1)-algebraic theory,
and some full -version?
]]>Are there any reasons to believe Kapranov over Theo (#17)?
I read Johnson-Freyd 15 as giving a neat reason why to consider the image of the J-homomorphism in the context that we are discussing. However Kapronov’s arguments for his suggestion (the super-representation categories and the appearance of for the string) lie outside that image. Still, both the arguments of Kapranov and of Johnson-Freyd seem to nicely fit together, I don’t see a tension between them regarding the picture that they suggest.
However presently neither helps me see what you ask for in #22 and what I said in #23 I am still stuck with: see what the -analog of supermanifolds such as the basic superpoints should be. An affine -scheme simply is -graded commutative in . We may choose to remember just the underlying -grading, and hence interpret this as an N-supermanifold, but the issue remains that the superpoints and superspacetimes of interest don’t lift to N-supermanifold structure.
On the other hand, maybe I am looking at it the wrong way somehow, which easily happens, as the idea we are after is so vague at the moment.
]]>Are there any reasons to believe Kapranov over Theo (#17)?
Moving up the ladder in representation theory, is there a reason that shares homotopy groups from 0 to 3 with ?
]]>I know the background, but there is a gap in the proposal that I myself made. I had said that the suggestion that super-grading is to be read as grading by the first or second stable homotopy groups of spheres, in turn suggests that super-geometry ought to be a shadow of -graded -geometry, and that this is particularly suggestive since that result by Sagave shows that every connective -ring is canonically -graded.
The gap in this suggestion of mine is that -grading taken at face value categorifies -grading (via ). And the induced -grading of any -ring is just the standard -grading of its .
Now it is true that the sign rule of this -grading comes from the underlying -grading after reduction mod 2, so that every -scheme has underlying it what some people, following Schwarz et al., call an “N-supermanifold” (or N-superscheme). But not every supermanifold is an N-supermanifold, and in particular the crucial ones like super-spacetimes are not. And finally, there are the arguments by Kapranov that the true supergrading (not the one arising as a shadow of cohomological gradiing) happens in and , not in .
All of this suggests that if we are to see genuine super-geometry as a shadow in -geometry, it should appear after some kind of looping. does not seem to be want to be a higher super-point. “” would seem to be what we want, only that it does not make sense, since is not an -ring.
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