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seeing Eric create diffeology I became annoyed by the poor state that the entry diffeological space was in. So I spent some minutes expanding and editing it. Still far from perfect, but a step in the right direction, I think.
(One day I should add details on how the various sites in use are equivalent to using CartSp)
Moving discussion here and summarizing content in the text
+– {: .query} Mike: Why only rings without units (that is, rngs)? Intuitively, what important properties do the above listed examples share that are not shared by rings with units?
Zoran Skoda: I want to know the answer as well. It might be something in the self-dual axioms. For unital rings artinian implies noetherian but not other way around; though the definitions of the two notions are dual.
Toby: The category of unital rings and unitary ring homomorphisms has no zero object.
Mike: Ah, right. Is it protomodular? I think I will understand this definition better from some non-examples that violate each clause individually.
walt: It is protomodular. This follows from the main theorem of Characterization of Protomodular Varieties of Universal Algebra by Bourn and Janelidze. By that theorem any variety that contains a group will be protomodular. Unital rings only fail to be semiabelian for the trivial reason that ideals aren’t subrings.
=–
Maybe the result on protomodularity (with citation) mentioned by walt citing Bourn and Janelidze should be moved to CRing (and also Ring, if it holds for non-commutative rings).
I have added a comment and collected some references on the renormalization freedom in the cosmological constant: here
I have cross-linked this with related entries: renormalization, perturbative quantum gravity and stress-energy tensor
brief page for one more entry in the brane scan, on occasion of today’s
I gave the brane scan table a genuine Lab incarnation and included it at Green-Schwarz action functional and at brane.
added pojnter to Maldacena-Nunez 01
Modulo the definition, I’ve created Picard scheme. One thing I couldn’t tell, is there a standard term in nlab for the “fiber category” of a stack? I mean if fibers over then if you pick some object from the category consisting of objects that go to and morphisms that go to .
I added some material to Mal’cev variety, namely proofs showing the various characterizations are equivalent, and a brief Examples section.
Just to highlight that in rev 2, from Oct. 2013 Xiao-Gang Wen added a remark claiming that the single reference given here is no good.
Wen added the analogous comment to topological insulator in rev 5, from Oct 2013
(I haven’t looked into it yet, just highlighting the edit for the moment, which seems to have gone unnoticed.)
brief category:people-entry for hyperlinking references at equivariant bundle
I have finally created a stub entry for universal principal bundle (which used to redirect to universal principal infinity-bundle ).
brief category:people-entry for hyperlinking references at equivariant principal bundle
brief category:people-entry for hyperlinking references at equivariant bundle
I changed back the name of the page to coherent state. Though it is usually considered in quantum mechanics, and the name is still correct, as a specialist in the area of coherent states, I have almost never seen the phrase “coherent quantum state” written out in mathematical physics, so I would prefer to have this long unusual name as a redirect only. Of course, we often talk about the coherence of quantum states. But this is about a general feature of coherence, like in optics. The specific states in mathematical physics which, among other features, have such coherence properties are usually called squeezed coherent states, and the coherent states of these entry are even more specific than those. I am about to add a couple of new references, so I came across the page again.
added a little bit of content to cohomology operation
brief category:people-entry for hyperlinking references at equivariant bundle
started a Properties-section at Lawvere theory with some basic propositions.
Would be thankful if some experts looked over this.
Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.
just for completeness, to go alongside structure group, gauge group, etc.
brief category:people-entry for hyperlinking references at spectrum (geometry) and at multi-adjoint
At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group together with the bundary map and the -action on . As someone confirms this example is correct I’ll add it to crossed module.
brief category:people-entry for hyperlinking references at spectrum (geometry) and now also at mutli-adjoint
at subobject classifier I have cleaned up the statement of the definition and then indicated the proof that in locally small categories subobject classifiers precisely represent the subobject-presheaf.
I added to field a mention of some other constructive variants of the definition, with a couple more references.
brief category:people-entry for hyperlinking references at twisted equivariant K-theory
brief category:people-entry for hyperlinking references at equivariant K-theory, twisted equivariant K-theory, orbifold K-theory and Verlinde ring
This needs an entry of its own (currently hidden in a subsection of anomalous magnetic moment).
Just starting something here, from my phone over coffee. More later.
this has been seen over for a while now; time to record some references and to relate to flavour anomaly.
Just starting here, from my phone over coffee. Nothing much to see here yet.
we had created limit, reflected limit, preserved limit, but not lifted limit. I have now created a stub for the last one, for completeness.
Would be good to harmonize and cross-relate these four entries more…
Stub. For the moment just for providing a place to record this reference:
brief category:people-entry for hyperlinking references at equivariant K-theory
added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:
On general grounds, since orbifolds are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces
into any desired coefficient ∞-stack (or sheaf of spectra) .
More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold
for a given -action on some manifold , should coincide with the -equivariant cohomology of . However, such an identification (1) is not unique: For any closed subgroup, we have
This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of global equivariant homotopy theory, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.
Concretely, in global equivariant homotopy theory the plain orbit category of -equivariant Bredon cohomology is replaced by the global orbit category whose objects are the delooping stacks , and then any orbifold becomes an (∞,1)-presheaf over by the evident “external Yoneda embedding”
More generally, this makes sense for any orbispace. In fact, as a construction of an (∞,1)-presheaf on it makes sense for any ∞-stack, but supposedly precisely if is an orbispace among all ∞-stacks does the cohomology of in the sense of global equivariant homotopy theory coincide the cohomology of in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.
At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)
brief category:people-entry for hyperlinking references at orbifold cohomology
added pointer to
splitting this off from Cohomotopy to make room for discussion of the Sullivan models from
As there had been a change to the entry for Ross Street I gave it a glance. Is there a reason that the second reference is to a paper without Ross as an author?I hesitate to delete it as there may be a hidden reason. (I have edited this discussion entry to remedy the point that Todd and Urs have made below. I also edited the title of this discussion!)
added pointer to
on relating the Freed-Witten anomaly to the shifted C-field flux quantization
(also linked the other way around)
brief category:people-entry for hyperlinking references at twisted differential K-theory
brief category:people-entry for hyperlinking references at twisted differential K-theory
added some lines on the relation to D-brane charge (spun off from additions I just made to twisted differential cohomoloy), and added more references
brief category:people-entry for hyperlinking references at twisted differential K-theory and at equivariant ordinary differential cohomology
brief category:people-entry for hyperlinking references at Witten-Sakai-Sugimoto model and at nuclear matrix model
brief category:people-entry for hyperlinking references at nuclear matrix model
brief category:people-entry for hyperlinking references at nuclear matrix model and elsewhere