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I created Dieudonne module. What is the policy on accents. Technically this should be written Dieudonné module everywhere. There is a redirect. I also defined this in the affine case, but I’m pretty sure if you replace “affine” by “flat” everything should work still.
started unitary representation of the super Poincaré group – the super-analog of unitary representation of the Poincaré group – so far mainly in order to record some references.
Also created a stub super Poincaré group
I wrote Specker sequence, a topic in computability theory that also has applications to constructivism.
I’ve started the page Cartier module.
created an entry Bohrification
started an entry twisted spin structure. So far the main point is to spell out the general abstract definition and notice that this is what Murray-Singer’s “spin gerbes” are models of.
I added remarks on Cauchy completion to the Properties-section both at proset and poset.
Also made more explicit at poset the relation to prosets.
I notice that at proset there is a huge discussion section. It would be nice if those involved could absorb into the main text whatever stable insight there is, and move the remaining discussion to the nForum here.
I created the page Witt Cohomology.
Todd has added to Grothendieck topos the statement and proof that any such is total and cototal (and I have added to adjoint functor theorem the statement that this implies that all (co)limit preserving functors between sheaf toposes have (right)left adjoints).
I notice that we should really merge Grothendieck topos with category of sheaves. But I don’t have the energy to do this now.
I edited adjoint functor theorem a bit: gave it an Idea-section and a References-section and, believe it or not, a toc.
Then I opened an Examples-section and filled in what I think is an instructive simple example: the right adjoint for a colimit preserving functor on a category of presheaves.
added statement and pointer to the proof of the gravitational stability of Minkowski spacetime
I made a stubby start at unitary irreps of the Poincare group, titled this way to save space. Very eager to get to the bottom of things; this subject can't be that hard.
Happened to notice a question at bicartesian closed category.
Question: don’t you need distributive bicartesian closed categories to interpret intuitionistic propositional logic? Consider the or-elimination rule
The intepretations of the two premises will be maps of type and . Then the universal property of coproducts gets us to , but we can’t get any farther – we need a distributivity law to get .
stub for Killing vector
for the moment, out of laziness, I also made Killing spinor and covariantly constant spinor redirect to this
I completed the proof of the corollary which states that for any monad on , that has colimits.
I’ve started a page on the height of a variety. This is something I’ll hopefully add a ton to later. It will probably require me to add pages on Dieudonne modules, p-divisible groups, and Witt cohomology at some point.
I have created an entry spinning particle
As you can see there, so far the only point this entry is making is that the worline action functional for the ordinary Dirac spinor (such as the electrons and quarks that we all consist of) happpens to be supersymmetric. I have written a little paragraph discussing this in words a little, and then mainly collected a list of references that explain this.
To be further expanded.
I have only now discovered that Gonzalo Reyes is (or has been) running a blog where he has posted lots of useful-looking notes.
For instance in the Physics-section he has a long series of expositions on basics of differential geometry with an eye towards general relativity in terms of synthetic differential geometry. I have added pointers to this to various related entries now.
I added the sentence
The factorizing morphism is sometimes called the corestriction of :
to image and made corestriction redirect to this page.
am adding references to Alexandrov space
The entry quantum state had been a bad mess with much dubious material. Where it was not dubious, it was superceded by the parallel state in AQFT and operator algebra.
For the time being I have mostly cleared this entry and added a pointer to state in AQFT and operator algebra. I think the best would be to delete the content of this entry entirely and merge the material from “state in AQFT and operator algebra” into here. But I am not energetic enough at this time of night to do so yet.
I have split off classical state as a separate entry, which was implicit in some other entries.
New stub Yuri Matiyasevich and additions to number theory aka arithmetic.
started the trinity of entries
But not done yet. So far: the basic idea in words and a pointer in each entry to the corresponding section in Zeidler’s textbook.
added a bit of substance to functional calculus
Another meaning at operator, and the connection between them.
At homomorphism, an incorrect definition was given (at least for monoids, and this was falsely claimed to generalise to the definition of functor). So I fixed this, and in the process expanded it (spelling out the inadequacy of the traditional definition for monoids) and made several examples (made explicit in the text) into redirects.
I have created a stub type II string theory, because I needed the link. Hopefully at some point I find the time to write something substantial about the classification of critical 2d SCFTs. But not right now.
started sewing constraint
I have added to perturbation theory and to AQFT a list of literature on perturbative constructions of local nets of observables.
This is in reply to a question Todd was asking: while the rigorous construction of non-perturbative interacting QFTs in dimension is still open, there has at least been considerable progress in grasping the perturbation theory and renormalization theory known from standard QFT textbooks in the precise context of AQFT.
This is a noteworthy step: for decades AQFT had been suffering from the lack of examples and lack of connection to the standard (albeit non-rigorous) literature.
Remake of Street’s Gummersbach paper: Characterization of Bicategories of Stacks (zoranskoda).
Urs, while it is good that spectral theorem is included into functional analysis table of contents, and it has functional analysis toc bar, I do not like that spectral theory is also included and also has this toc bar. My understanding is that spectral theory is much wider subject on the relation between the possibly categorified and possibly noncommutative function spaces (sheaf categories, noncommutative analogues) and the specifical “singular” features of those like prime ideals, like certain special objects in abelian categories, points of spectra in operator framework etc. In any case, in POV, it is NOT a part of functional analysis, though some manifestations are. Like the concept of a space is not a subject of functional analysis, though some spaces are defined in the language of operator algebras. I find spectral theory on equal footing like space, “quantity” etc. Of course, the entry currently does not reflect this much (though it has a section on spectra in algebraic geometry), but it eventually will! Thus I will remove it from functional analysis contents.
One should also point out that using generators in the proof of Giraud’s reconstruction theorem of a site out of a topos is a variant of spectral idea: like points form certain spaces, so the generators of various kind generate or form a category. This is behind many spectral constructions (including recent Orlov’s spectrum which is very laconic but stems from that) and reconstruction theorems and if the category corresponds to coherent sheaves over a variety than often the geometric features of the variety give certain contributions to the spectrum.
Can someone look at Three Roles of Quantum Field Theory. There was an unsigned change there and a box that does not work. I do not know what was intended so will not try to fix it.
New entry Dmytro Shklyarov; he seems to be now in Augusburg. Lots of interesting recent work in several subfields of our interest. I did not know where to put his 2-representations paper into 2-vector space as the bibliography is scattered there with some classification of subtopics.
At Urs’ urging, I have created functional analysis - contents. It needs considerable extending; and I’ve yet to include it anywhere.
As hinted by the contents, I plan to move the diagram from TVS to its own page (but still include it on TVS).
Created entry Hall algebra with a list of references and links for now. Related name entries, Daniel Huybrechts, Bernhard Keller, and updates to Berntrand Toen (and for the heck, Bernhard Riemann), and to contributors to algebraic geometry.
I wrote about these at measurable space, following to reference to M.O answers by Dmitri Pavlov that were already being cited.
have a look into (the) future
I have split off E-infinity operad from little k-cubes operad (where it had been hiding well) and expanded a bit
(in reply to a question by John over on Azimuth)
I have added some new material to Boolean algebra and to ultrafilter. In the former, I coined the term ’unbiased Boolean algebra’ for the notion which describes Boolean algebras as equivalent to finite-product-preserving functors from the category of finite nonempty sets, and the term -biased Boolean algebra to refer to the multiplicity of ways in which Boolean algebras could be considered monadic over .
In ultrafilter, I added some material which gives a number of universal descriptions of the ultrafilter monad. This is in part inspired by some discussions I’m having with Tom Leinster, who remarked recently at the categories list that the ultrafilter monad could be described as a codensity monad. All this is related to the unbiased Boolean algebras and to the remarks due to Lawvere, which were described on an earlier revision; this material has been reworked.
created an entry (infinity,1)Toposes on the -catgeory (or -category) of all -toposes.
Also split off an entry (infinity,1)-geometric morphism
I have tried to make the page torsion look more like a disambiguation page and less like a mess. But only partially successful.
stub for quasi-state
stub for Wigner’s theorem
I have split off from smooth infinity-groupoid – structures the section on concrete objects, creating a new entry concrete smooth infinity-groupoid.
Right now there is
a proof that 0-truncated concrete smooth -groupoids are equivalent to diffeological spaces;
and an argument that 1-truncated concrete smooth -groupoids are equivalent to “diffeological groupoids”: groupoids internal to diffeological spaces.
That last one may require some polishing.
I am still not exactly sure where this is headed, in that: what the deep theorems about these objects should be. For the moment the statement just is: there is a way to say “diffeological groupoid” using just very ygeneral nonsense.
But I am experimenting on this subject with Dave Carchedi and I’ll play around in the entry to see what happens.
I thought about starting a floating toc for classifying objects and related, but then decided to subsume it into Yoneda lemma - contents. There I have now added the list of entries
and, conversely, included that toc into all these entries.
have split off the definition of umbrella category from subterminal object
since the link was requested somewhere, I have created a stub for n-topos
In convenient category of topological spaces, I rewrote a little under the section on counterexamples, and I added a number of examples and references. Some of this came about through a useful exchange with Alex Simpson at MO, here.
I got a question by email about the equivariant tubular neighbourhoods in loop spaces (as opposed to those defined using propagating flows so I figured it was time to nLabify that section of differential topology of mapping spaces. Of course, in so doing I figured out a generalisation: given a fibre bundle , everything compact, we consider smooth maps which are constant on fibres. This is a submanifold of the space of all smooth maps . Assuming we can put a suitable measure on the fibres of , then we can define a tubular neighbourhood of this submanifold.
Details at equivariant tubular neighbourhoods. Title may be a bit off now, but it’s that because the original case was for the fibre bundle with fibre .
This entry is also notable because I produced it using a whole new LaTeX-to-iTeX converter. Details on the relevant thread.
I added a reference to a paper of Connes and Rovelli (1994) and a link (in modular theory) to
where André Henriques asks about some Connes philosophy. But André quotes in explaining the background to his question, that in full generality there is a homomorphism from imaginary line into the 2-group of invertible bimodules of the given von Neumann algebra , which in the presence of state lifts to the homomorphism into . I learned just the case when there is a state, and am delighted to hear that this is just a strengthening of some categorical structure which exists even more generally. If somebody is familiar or can dig more on that general case, it would be nice to have such categorical picture in the Lab entry modular theory.
Specially good for usage and references in our foundational entries on quantum mechanics is that they have excellent online articles quantum logic and probability theory, quantum mechanics: Kochen-Specker theorem, quantum mechanics and quantum mechanics: von Neumann vs. Dirac.
Igor Bakovic created Diaconescu’s theorem
you may recall (okay, probably not ;-) what I once wrote in the entry on exterior differential systems: while in the classical literature these are thought of as dg-ideals in a de Rham complex, we should think of them as sub-Lie algebroids of tangent Lie algebroids.
Since exterior differential systems over X encode and are encoded by partial differential equations on functions on X, this means that such sub-Lie algebroids are partial differential equations.
This perspective is amplified much more in the literature on D-modules: I think we can think of a D-scheme as an infinite-order analog of a Lie algebroid, which is the corresponding first-order notion. The Jet-bundle with its D-scheme structure is the infinite-order analog of the tangent Lie algebroid.
And sub-D-schemes of Jet-D-schemes are partial differential equations, this is what everyone on D-geometry tells you first.
So I think there is a nice story here.
New entry affiliated operator of a -algebra aka affiliated element. This is important for the circle of entries on algebraic QFT, as the operator algebras are formed by bounded operators, while we typically need unbounded operators like derivative operator to do quantum mechanics.
I sent a version of that entry but the Lab stuck in the middle of the operation so I am not sure if I succeeded. So here is the copy:
Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on which is proportional to the momentum operator in quantum mechanics.
The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of “approximable” operators by means of operator algebra itself. One way to do this is to define the affiliated elements of -algebra, or the operators affiliated with the -algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the spectral decomposition will supply the approximation.