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started Brauer group, collecting some references on the statement that/when Br(X)≃H2et(X,𝔾m)tor and moved notes from a talk by David Gepner on ∞-Brauer groups to there.
Created:
Maharam’s theorem states a complete classification of isomorphism classes of the appropriate category of measurable spaces.
In the σ-finite case, the theorem classifies measure spaces up to an isomorphism. Here an isomorphism is an equivalence class of measurable bijections f with measurable inverse such that f and f−1 preserve measure 0 sets.
As explained in the article categories of measure theory, for a truly general, unrestricted statement for non-σ-finite spaces there are additional subtleties to consider: equality almost everywhere must be refined to weak equality almost everywhere, and σ-finiteness should be relaxed to a combination of Marczewski-compactness and strict localizibility.
In this unrestricted form, by the Gelfand-type duality for commutative von Neumann algebras, Maharam’s theorem also classifies isomorphism classes of localizable Boolean algebras, abelian von Neumann algebras, and hyperstonean spaces (or hyperstonean locales).
Every object in one of the above equivalent categories canonically decomposes as a coproduct (disjoint union) of ergodic objects. Here an object X is ergodic if the only subobjects of X invariant under all automorphisms of X are ∅ and X itself.
Furthermore, an ergodic object X is (noncanically, using the axiom of choice) isomorphic to 𝔠×2κ, where κ is 0 or infinite, and 𝔠 is infinite if κ is infinite. Here the cardinal 𝔠 is known as the cellularity of X and κ is its Maharam type.
In particular, if κ=0, we get a classification of isomorphism classes of atomic measure spaces: they are classified by the cardinality 𝔠 of their set of atoms.
Otherwise, κ is infinite, and we get a classification of isomorphism classes of ergodic atomless (or diffuse) measure spaces: such spaces are isomorphic to 𝔠×2κ, where 𝔠 and κ are infinite cardinals.
Thus, a completely general object X has the form
∐κ𝔠κ×2κ,where κ runs over 0 and all infinite cardinals, 𝔠κ is a cardinal that is infinite or 0 if κ≠0, and 𝔠κ≠0 only for a set of κ.
The original reference is
A modern exposition can be found in Chapter 33 (Volume 3, Part I) of
I created Bishop’s constructive mathematics by moving some material from Errett Bishop and adding some more discussion of what it is and isn’t. Comments and suggestions are very welcome; I’m still trying to figure out the best way to describe the relationship of this theory to other things like topos logic.
I am starting something at six operations.
(Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)
Have added more of the original (“historical”) References with brief comments and further pointers.
(not an edit but to create the forum thread) Is the characterization in As an 11-dimensional boundary condition for the M2-brane complete or does one need to further extend by the m5 cocycle?
am starting differential string structure, but not much there yet
I have added pointer to
to the entries 7-sphere, ADE classification, Freund-Rubin compactification.
This article proves the neat result that the finite subgroups Γ of SO(8) such that S7/Γ is smooth and spin and has at least four Killing spinors has an ADE classification. The Γs are the the “binary” versions of the symmetries of the Platonic solids.
For the purposes of negative thinking, it may be useful to recognise that every ∞-category has a (−1)-morphism, which is the source and target of every object. (In the geometric picture, this comes as the (−1)-simplex of an augmented simplicial set.)
Jonathan Arnoult has pointed out on CT Zulip that this is misleading: it sounds like it implies that every ∞-category is monoidal! And John Baez pointed out that the analogy to augmented simplicial sets fails because in an augmented simplicial set each 0-simplex has only one face, rather than a separate “source” and “target” that are both the same (−1)-simplex.
I suggest we just remove this paragraph and the query box following it, since I can’t think of a way to rephrase it that would be more helpful than unhelpful. But I’m open to other suggestions.
Stub. For the moment just for providing a place to record this reference:
This is a brief description of the construction that started appearing in category-theoretic accounts of deep learning and game theory. It appeared first in Backprop As Functor (https://arxiv.org/abs/1711.10455) in a specialised form, but has slowly been generalised and became a cornerstone of approaches unifying deep learning and game theory (Towards Foundations of categorical Cybernetics, https://arxiv.org/abs/2105.06332), (Categorical Foundations of Gradient-based Learning, https://arxiv.org/abs/2103.01931).
Our group here in Glasgow is using this quite heavily, so since I couldn’t find any related constructions on the nLab I decided to add it. This is also my first submission. I’ve read the “HowTo” page, followed the instructions, and I hope everything looks okay.
There’s quite a few interesting properties of Para, and eventually I hope to add them (most notably, it’s an Para is an oplax colimit of a functor BM -> Cat, where B is the delooping of a monoidal category M).
A notable thing to mention is that I’ve added some animated GIF’s of this construction. Animating categorical concepts is something I’ve been using as a pedagogical tool quite a bit (more here https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html) and it seems to be a useful tool getting the idea across with less friction. If it renders well (it seems to) and is okay with you, I might add more to the Optics section, and to the neural networks section (I’m hoping to get some time to add our results there).
Bruno Gavranović
added pointer to:
added working institute webpage link
as well as
created a “category: reference”-page The Stacks Project
I have only now had a closer look at this and am impressed by the scope this has. Currently a total of 2288 pages. It starts with all the basics, category theory, commutative algebra and works its way through all the details to arrive at algebraic stacks.
So besides my usual complaint (Why behave as if there are not sites besides the usual suspects on CRingop and either give a general account or call this The Algebraic Stacks Project ? ) I am enjoying seeing this. We should have lots of occasion to link to this. Too bad that this did not start out as a wiki.
Added this reference
I needed an entry to be able to point to which collects pointers to the various entries on “dualities” in string theory. So I created one: duality in string theory.
I have edited at Tychonoff theorem:
tidied up the Idea-section. (Previously there was a long paragraph on the spelling of the theorem before the content of the theorem was even mentioned)
moved the proofs into a subsection “Proofs”, and added a pointer to an elementary proof of the finitary version, here
Notice that there is an ancient query box in the entry, with discussion between Todd and Toby. It would be good to remove this box and turn whatever conclusion was reached into a proper part of the entry.
At then end of the entry there is a line:
More details to appear at Tychonoff theorem for locales
which however has not “appeared” yet.
But since the page is not called “Tychonoff theorem for topological spaces”, and since it already talks about locales a fair bit in the Idea section, I suggest to remove that line and to simply add all discussion of localic Tychonoff to this same entry.
Started lift.
weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.
Would it be better to have these redirect to lift?
added pointer to today’s
created quick stub for framed bicategory
but my machine's battery will die any second now...
I have added to string theory a new section Critical strings and quantum anomalies.
Really I was beginning to work on a new entry twisted spin^c structure (not done yet) and then I found that a summary discussion along the above lines had been missing.
I added a reference on Gabriel filters on quantales.
Very strange: version one on show had a reference on Etendues (entered by T. Holder), but when I clicked edit there was none in the edit window, just in show window. I edited and one can not see it in any history, nowhere. So there was something in show cache from 2014 which is not recorded in any history edit.
I gave Fourier-Mukai transform a bit of an Idea-section. It overlaps substantially with the Definition section now, but I thought one needs to say the simple basic idea clearly in words first. Also added a few more pointers to literature.
I am giving Modern foundations for stable homotopy theory a category:reference entry.
First thing I did was to brush-up the list of references at symmetric smash product of spectra. Then I copied over the nicely to-the-point History-paragraph to a new section stable homotopy theory – history.
have now spelled out at Tor in simple terms how TorAb1(A,B) is a torsion group, so far for the case that A is finite.
added to equalizer statement and proof that a category has equalizers if it has pullbcks and products
I have added
and added publication details to
and grouped together more discernibly the references on operator-algebraic entropy