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added this:
Relation to (identification with) much older inequalities in classical probability theory, due to George Boole was pointed out by (among others, called the “probabilistic opposition” in Khrennikov 2007, p. 3):
Luigi Accardi, The Probabilistic Roots of the Quantum Mechanical Paradoxes, in: The Wave-Particle Dualism, Fundamental Theories of Physics 3 Springer (1984) [doi:10.1007/978-94-009-6286-6_16
Itamar Pitowsky, From George Boole To John Bell — The Origins of Bell’s Inequality, in: Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Fundamental Theories of Physics 37 Springer (1989) [doi:10.1007/978-94-017-0849-4_6]
reviewed in
Elemer E Rosinger, George Boole and the Bell inequalities [arXiv:quant-ph/0406004]
{#Khrennikov07} Andrei Khrennikov, Bell’s inequality: Physics meets Probability [arXiv:0709.3909]
Andrei Khrennikov, Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?, Entropy 10 2 (2008) 19-32 [doi:10.3390/entropy-e10020019]
Created:
An analogue of the Gelfand duality for commutative von Neumann algebras.
The following five categories are equivalent.
The opposite category of commutative von Neumann algebras and normal *-homomorphisms.
The category of measurable locales, which is a full subcategory of the category of locales.
The category of hyperstonean locales and open maps of locales.
The category of hyperstonean spaces and open continuous maps.
The category of compact strictly localizable enhanced measurable spaces and equivalence classes of measurable negligibility-reflecting maps modulo the equivalence relation of weak equality almost everywhere.
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.
have started something at orthosymplectic super Lie algebra and have added little bits and pieces to various related entries, such as first sketchy notes at super Lie algebra – classification and at supersymmetry – Classification – superconformal symmetry.
Nothing of this is done yet, but I need to call it quits now.
added a few references and links to super Lie algebra
at cyclic group there had been a typo that said “free group” instead of “cyclic group” (in the Examples-section). I have fixed that.
Somehow I think this entry could be organized differently, but I won’t do that now.
I just discovered that, all along, the term “quiver representation” was just redirecting to representation. Have started this dedicated page now, with the bare minimum
Some tidying up and additions at simplex category, in particular a section on its 2-categorical structure, and more on universal properties.
I’ve edited the definition to focus more on the augmented simplex category instead of the ’topologists’ ’, but I haven’t changed their names, because it seemed to me that that was the best way to keep everyone involved in the discussion at that page happy. (I also changed the ordinal sum functor from to , after Tim’s suggestion.)
stub for Calabi-Yau algebra
Added a diagram to cone and changed some notation to be compatible with cone morphism and Understanding Constructions in Set
Following discussion in some other threads, I thought one should make it explicit and so I created an entry
Currently this contains some (hopefully) evident remarks of what “dependent linear type theory” reasonably should be at least, namely a hyperdoctrine with values in linear type theories.
The entry keeps saying “should”. I’d ask readers to please either point to previous proposals for what “linear dependent type theory” is/should be, or criticise or else further expand/refine what hopefully are the obvious definitions.
This is hopefully uncontroversial and should be regarded an obvious triviality. But it seems it might be one of those hidden trivialities which deserve to be highlighted a bit more. I am getting the impression that there is a big story hiding here.
Thanks for whatever input you might have.
edited Moore closure. Added a bit more glue, restructured slightly, and added more hyperlinks.
reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.
I am beginning to give the entry FQFT a comprehensive Exposition and Introduction section.
So far I have filled some genuine content into the first subsection Quantum mechanics in Schrödinger picture.
But I have to quit now. This isn’t even proof-read yet. So don’t look at it unless you feel more in editing-mood than in pure-reading-mood.
cross-linked with super Klein geometry
added to the Properties-section at Hopf algebra a brief remark on their interpretation as 3-vector spaces.
stub for von Neumann algebra factor
I am starting an entry spontaneously broken symmetry. But so far no conceptualization or anything, just the most basic example for sponatenously broken global symmetry.
I added to star-autonomous category a mention of “-autonomous functors”.
I am moving the following old query box exchange from orbifold to here.
old query box discussion:
I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.
Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?
Urs Schreiber: please, go ahead. It would be appreciated.
end of old query box discussion
The new second edition is recorded at Practical Foundations for Programming Languages plus a link to a description of the changes.
crated D'Auria-Fre formulation of supergravity
there is a blog entry to go with this here
added to supergeometry a link to the recent talk
finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).
added a few more references with brief comments to QFT with defects
(this entry is still just a stub)
while adding to representable functor a pointer to representable morphism of stacks I noticed a leftover discussion box that had still be sitting there. So hereby I am moving that from there to here:
[ begin forwarded discussion ]
+–{+ .query} I am pretty unhappy that all entries related to limits, colimits and representable things at nlab say that the limit, colimit and representing functors are what normally in strict treatment are just the vertices of the corresponding universal construction. A representable functor is not a functor which is naturally isomorphic to Hom(-,c) but a pair of an object and such isomorphism! Similarly limit is the synonym for limiting cone (= universal cone), not just its vertex. Because if it were most of usages and theorems would not be true. For example, the notion and usage of creating limits under a functor, includes the words about the behaviour of the arrow under the functor, not only of the vertex. Definitions should be the collections of the data and one has to distinguish if the existence is really existence or in fact a part of the structure.–Zoran
Mike: I disagree (partly). First of all, a functor equipped with an isomorphism is not a representable functor, it is a represented functor, or a functor equipped with a representation. A representable functor is one that is “able” to be represented, or admits a representation.
Second, the page limit says “a limit of a diagram … is an object of equipped with morphisms to the objects for all …” (emphasis added). It doesn’t say “such that there exist” morphisms. (Prior to today, it defined a limit to be a universal cone.) It is true that one frequently speaks of “the limit” as being the vertex, but this is an abuse of language no worse than other abuses that are common and convenient throughout mathematics (e.g. “let be a group” rather than “let be a group”). If there are any definitions you find that are wrong (e.g. that say “such that there exists” rather than “equipped with”), please correct them! (Thanks to your post, I just discovered that Kan extension was wrong, and corrected it.)
Zoran Skoda I fully agree, Mike that “equipped with” is just a synonym of a “pair”. But look at entry for limit for example, and it is clear there that the limiting cone/universal cone and limit are clearly distinguished there and the term limit is used just for the vertex there. Unlike for limits where up to economy nobody doubt that it is a pair, you are right that many including the very MacLane representable take as existence, but then they really use term “representation” for the whole pair. Practical mathematicians are either sloppy in writing or really mean a pair for representable. Australians and MacLane use indeed word representation for the whole thing, but practical mathematicians (example: algebraic geometers) are not even aware of term “representation” in that sense, and I would side with them. Let us leave as it is for representable, but I do not believe I will ever use term “representation” in such a sense. For limit, colimit let us talk about pairs: I am perfectly happy with word “equipped” as you suggest.
Mike: I’m not sure what your point is about limits. The definition at the beginning very clearly uses the words “equipped with.” Later on in the page, the word “limit” is used to refer to the vertex, but this is just the common abuse of language.
Regarding representable functors, since representations are unique up to unique isomorphism when they exist, it really doesn’t matter whether “representable functor” means “functor such that there exists an isomorphism ” or “functor equipped with an isomorphism .” (As long as it doesn’t mean something stupid like “functor equipped with an object such that there exists an isomorphism .”) In the language of stuff, structure, property, we can say that the Yoneda embedding is fully faithful, so that “being representable” is really a property, rather than structure, on a functor.
[ continued in next comment ]
Added Clark’s comment that many of their bases are EI (∞,1)-categories. I guess many of these are in addition inverse EI (∞,1)-categories.
Is a stratification always well-founded?