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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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)

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• 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).

• Added link to Bourn’s most helpful 2017 textbook From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories. Revised reference to the Borceux-Bourn 2004 monograph.

• a brief stub, to make links work

• a bare list of references, to be !include-ed into the References-list of relevant entries

• 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 $F:C\to D$ fibers $C$ over $D$ then if you pick some object $X$ from $D$ the category $C_X$ consisting of objects that go to $X$ and morphisms that go to $id_X$.

• I added some material to Mal’cev variety, namely proofs showing the various characterizations are equivalent, and a brief Examples section.

• Created.

• 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

• 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.

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Anonymous

• 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.

• Created page.

• At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group $\pi_2(X,A)$ together with the bundary map $\delta:\pi_2(X,A)\to \pi_1(A)$ and the $\pi_1(A)$-action on $\pi_2(X,A)$. As someone confirms this example is correct I’ll add it to crossed module.

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• Added a reference to Osmond’s paper.

• 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.

• 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 $4\sigma$ 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.

• Stub. For the moment just for providing a place to record this reference:

• Jean Thierry-Mieg, Connections between physics, mathematics and deep learning, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)
• 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 $\mathcal{G}$ are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces

$H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )$

into any desired coefficient ∞-stack (or sheaf of spectra) $E$.

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

$\mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)$

for a given $G$-action on some manifold $X$, should coincide with the $G$-equivariant cohomology of $X$. However, such an identification (1) is not unique: For $G \subset K$ any closed subgroup, we have

$X \sslash G \;\simeq\; \big( X \times_G K\big) \sslash K \,.$

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 $Orb_G$ of $G$-equivariant Bredon cohomology is replaced by the global orbit category $Orb_{glb}$ whose objects are the delooping stacks $\mathbf{B}G \coloneqq \ast\sslash G$, and then any orbifold $\mathcal{G}$ becomes an (∞,1)-presheaf $y \mathcal{G}$ over $Orb_{glb}$ by the evident “external Yoneda embedding

$y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.$

More generally, this makes sense for $\mathcal{G}$ any orbispace. In fact, as a construction of an (∞,1)-presheaf on $Orb_{glb}$ it makes sense for $\mathcal{G}$ any ∞-stack, but supposedly precisely if $\mathcal{G}$ is an orbispace among all ∞-stacks does the cohomology of $y \mathcal{G}$ in the sense of global equivariant homotopy theory coincide the cohomology of $\mathcal{G}$ 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)

• Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf)
• splitting this off from Cohomotopy to make room for discussion of the Sullivan models from

• Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)
• starting something

• 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!)

• started adding something (the example of the Hopf fibration and some references).

What’s a canonical reference on the Whitehead products corresponding to the Hopf fibrations? Like what is an original reference and what is a textbook account?

• 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

• a bare list of entries, for ease of cross-linking these in their lists of Related entries

• brief category:people-entry for hyperlinking references at nuclear matrix model and elsewhere

• starting some minimum on Hashimoto et al.’s D4-brane matrix model for baryons/nuclei in the WSS model for QCD

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