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quick note at spin structure on the characterization over Kähler manifolds
Added another reference.
I was chatting with Robin Cockett yesterday at SYCO1. In a talk Robin claims to be after
The algebraic/categorical foundations for differential calculus and differential geometry.
It would be good to see how this approach compares with differential cohesive HoTT.
Changes made only to the Universal property of the 2-category of spans section. The citations by Urs lead to another citation which, in turn, leads to another citation. With a little effort, I tracked down the a full copy of said universal property, I’ve replicated it here, added the citation used, although I left the previous citations there for convenience; a more experienced editor can remove those if they would like.
I would like to note that the author whose work I have referenced, Hermida, also notes: “[this universal property] is folklore although we know no references for it.”
Please make any corrections needed and clean up the language here; this is a fairly direct copy of what is written, but I imagine somebody with more knowledge of all the language used here can rewrite this universal property stuff in a cleaner way.
Thanks!
Anonymous
at quantum observable there used to be just the definition of geometric prequantum observables. I have added a tad more.
I have expanded the Idea-section at 3d quantum gravity and reorganized the remaining material slightly.
I feel unsure about the pointer to “group field theory” in the References. Can anyone list results that have come out of group field theory that are relevant here?
I find the following noteworthy, and I am not sure if this is widely appreciated:
the original discussion of the quantization of 3d gravity by Witten in 1988 happens work out to be precisely along the lines that “loop quantum gravity” once set out to get to work in higher dimensions: one realizes
that the configuration space is equivalently a space of connections;
that these can be characterized by their parallel transport along paths in base space;
that therefore observables of the theory are given by evaluating on choices of paths (an idea that goes by the unfortunate name “spin network”).
All this is in Witten’s 1988 article. Of course the point there is that in the case of 3d this can actually be made to work. The reason is that in this case it is sufficient to restrict to flat connections and for these everything drastically simplifies: their parallel transport depends not on the actual paths but just on their homotopy class, rel boundary. Accordingly the “spin networks” reduce to evaluations on generators of the fundamental group, etc.
Notice that in 4d the analog of this step that Witten easily performs in 3d was never carried out: instead, because it seemed to hard, the LQG literature always passes to a different system, where smooth connections are replaced by parallel transport that is required to be neigher smooth nor in fact continuous. These are called “generalized connections” in the LQG literature. Of course these have nothing much to do with Einstein-gravity: because there the configuration space does not contain such “generalized” fields.
For these reasons I feel a bit uneasy when the entry refers to LQG or spin foams as “other approaches” to discuss 3d quantum gravity. First of all, the existing good discussion by Witten did realize the LQG idea already in that dimension, and it did it correctly. So in which sense are there “other approaches”?
Which insights on 3d quantum gravity do “spin foam”s or does “group field theory”add? If anyone could list some results with concrete pointers to the literature, I’d be most grateful.
I have half-heartedly started adding something to Kac-Moody algebra. Mostly refrences so far. But I don’t have the time right now to do any more.
starting something (the entry title is to rhyme with restricted representation and induced representation)
Unfortunately, there are two entries on the same topic, both created by Urs: quantum Hall effect (redirecting also fractional quantum Hall effect what should eventually split off) with some substance, and the microstub quantum hall effect. I would like to create quantum spin Hall effect and I think I should rename/reclaim the stub quantum hall effect for this. Do others agree ? Urs ?
As the action is now delayed I record here the reference which I wanted to put there
Somewhat surprisingly, the authors and roughly this work of them are mentioned (though not in the list of references) in a paper in algebraic geometry
which considers the mirror symmetry and topological states of matters (topological insulators in particular) as main applications.
started some minimum at exceptional field theory (the formulation of 11d supergravity that makes the exceptional U-duality symmetry manifest)
gave representation theory a little Idea-section, then added some words on its incarnation as homotopy type theory in context/in the slice over and added the following homotopy type representation theory – table, which I am also including in other relevant entries:
One more item for the list of Sullivan models – examples
added a bunch of pointers to the literature (with brief comments) at string scattering amplitude.
Also added a corresponding paragraph at effective field theory.
(this is still in reaction to that MO discussion, specifically to the question here)
I edited Trimble n-category:
added table of contents
added hyperlinks
moved the query boxes that seemed to contain closed discussion to the bottom. I kept the query box where I ask for a section about category theory for Trimble n-categories, but maybe we want to remove that, too. Todd has more on this on his personal web.
Added a link to the page on hypersheaves (which was expanded today).
fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.
After Urs’ post at the café about “Tricategory of conformal nets” at Oberwolfach I took a look at the paper Conformal nets and local field theory and noted that I would have to ask some trivial and boring questions about nomenclature before I could even try to get to the content.
One example is about “Haag duality”: It seems to me that we need a generalization of net index sets on the nLab that includes the bounded open sets used for the Haag-Kastler vacuum representation and the index sets used in the mentioned paper. One of the concept needed would be “causal index set”:
A relation on an index set (poset) is called a causal disjointness relation (and are called causally disjoint if ) if the following properties are satisfied:
(i) is symmetric
(ii) and implies
(iii) if is bounded from above, then for all implies .
(iv) for every there is a with
A poset with such a relation is called a causal index set.
Well, that’s not completly true, because in the literature that I know there is the additionally assumtion that contains an infinite unbounded sequence and hence is not finite (that whould be a poset that is ? what? unbounded?), that is not a condition imposed on posets on the nLab.
After this definition one can go on and define “causal complement”, the “causality condition” for a net and then several notions of duality with respect to causal complements etc. all without reference to Minkowski space or any Lorentzian manifolds.
Should I create a page causal index set or is there something similar on the nLab already that I overlooked?
category: people page for
Canadian bacon
I created Galois module. I also added further references to p-divisible group; in particular section 4.2 of Lurie’s survey of elliptic cohomology gives some generalization of the classical theory. I started also a page with -the somehow unfortunate- title relations of certain classes of group schemes- I intended it to give an overview and examples of the basic kinds of group schemes occurring in classical (algebraic) number theory (the page contains more or less two specific examples; so there is still development potential).
added pointer to:
I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.
added pointer to
and rewrote the Idea-section to make it clear that these authors require not just existence of left and right adjoints, but in fact an ambidextrous adjoint and satisfying an extract coherence condition.
Added a recent reference on Peirce’s Gamma graphs for modal logic. This describes his first approach via broken cuts rather than the later tinctured sheet approach. I keep meaning to see if there’s anything in the latter close to LSR 2-category of modes approach.
According to the broken-cut method, possibility is broken cut surrounding solid cut, while necessity is solid cut surrounding broken cut. Since solid cut is negation, broken cut signifies not-necessarily. Easy to see as the same pattern of three cuts, etc.
In the Alpha case, we’re to think of negated propositions as though written elsewhere on another sheet (or the back of the sheet). There seems to be a three-dimensionality to the graphs, e.g., the conditional as like a tube from one sheet to another, Wikipedia. I gather his later ideas on tinctured graphs had this idea of being inscribed on different sheets.
added pointer to:
in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:
Quantum mechanical systems
States and observables
Spaces of states
Flows and time evolution
Still incomplete and rough. But I have to quit now.
I added more to idempotent monad, in particular fixing a mistake that had been on there a long time (on the associated idempotent monad). I had wanted to give an example that addresses Mike’s query box at the bottom, but before going further, I wanted to track down the reference of Joyal-Tierney, or perhaps have someone like Zoran fill in some material on classical descent theory for commutative algebras (he wrote an MO answer about this once) to illustrate the associated idempotent monad.
Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.
I gave root of unity its own entry (it used to redirect to root), copied over the paragraph on properties of roots of unities in fields, and added a paragraph on the arithmetic geometry description via and across-pointer with Kummer sequence.
the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.
I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).
(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)