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

• I added the HoTT introduction rule for ’the’, then added a speculative remark on why say things like

The Duck-billed Platypus is a primitive mammal that lives in Australia.

• Added Todd's definition of *-quantale to quantale. Is there anything about enrichment in such things that's worth adding?
• I added the description of fibrations for Dwyer-Kan model structure and stated that it’s cofibrantly generated (proof in Tabuada’s paper).

Polydarya

• First mention of coflare forms on the Lab rather than the Forum.

• I added the suggestion that there is a version for HoTT.

• I got rid of the old wedge products of $\mathrm{d}s$, which never worked out, and put $\mathrm{d}S$ in terms of $\mathrm{d}\mathbf{x}$ rather than $\mathrm{d}s$.

• created an extemely stubby stub Weiss topology, just to record pointer to that cool fact which Dmitri Pavlov advertised on MO (here).

I have no time to expand on the entry right now. But maybe somebody else here does? Would be worthwhile.

• Add some notes about getting twelf to work on modern Ubuntu

• fixed the statement of Example 5.2 (this example) by restricting it to $\mathcal{C} = sSet$

• added to axion a pointer to this recent article:

• Joseph P. Conlon, M.C. David Marsh, Searching for a 0.1-1 keV Cosmic Axion Background (arXiv:1305.3603)

Primordial decays of string theory moduli at $z \sim 10^{12}$ naturally generate a dark radiation Cosmic Axion Background (CAB) with $0.1 - 1 keV$ energies. This CAB can be detected through axion-photon conversion in astrophysical magnetic fields to give quasi-thermal excesses in the extreme ultraviolet and soft X-ray bands. Substantial and observable luminosities may be generated even for axion-photon couplings $\ll 10^{-11} GeV^{-1}$. We propose that axion-photon conversion may explain the observed excess emission of soft X-rays from galaxy clusters, and may also contribute to the diffuse unresolved cosmic X-ray background. We list a number of correlated predictions of the scenario.

• At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

• A stub.

• Looking back at an old Café thread, I see Neil Strickland telling us about Baas-Sullivan theory.

1) Baas-Sullivan theory allows you to start with a cobordism spectrum R and introduce singularities to construct R-module spectra that can be thought of as R/(x1,…,xn), where xi ∈ π*R.

2) This is computationally tractable when the elements xi form a regular sequence. You can construct connective Morava K-theories from complex cobordism this way, for example. You can also get ordinary homology, as the cobordism theory of complexes that are allowed arbitrary singularities of codimension at least two.

3) The original Baas-Sullivan framework is quite technical, and combinatorially complex. It is now easier to use the framework developed in the book by Elmendorf, Kriz, Mandell and May.

4) This procedure always gives R-modules, so if you start with MU (= complex cobordism) or MSO or MO, you will always end up with something complex orientable. In particular, you cannot get tmf or KO or the sphere spectrum from MU.

5) You can get more things if you do cobordism of manifolds with extra structure, such as a spin bundle or framing, for example. It is probably possible to get kO from MSpin. It might even be possible to get tmf from MString.

6) There is a theorem that I think appears in an old book by Buoncristiano, Rourke and Sanderson, showing that any generalised homology theory is a cobordism theory of manifolds with some kind of extra structure and singularities. I don’t think that they were able to given any nonobvious concrete examples other than ordinary homology, and I don’t think that anyone else has managed to go anywhere with this theory. But perhaps it would be worth taking another look.

Let’s see if any passing expert can help with an entry.

• I started a new page Yoneda structure. For the moment mostly references though hopefully I will supply more content in the coming days.

• Someone anonymous has noted that the labels in two diagrams in triangle identities are misplaced. This seems clear. As the diagrams are external, can someone edit them who has access to the original code? There seem to be other errors (e.g. a C should be a D), as well.