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• for completeness

• a minimum, to give a home to

• I do not understand the entry G-structure. G-structure is, as usual, defined there as the principal $G$-subbundle of the frame bundle which is a $GL(n)$-principal bundle. I guess this makes sense for equivariant injections along any Lie group homomorphism $G\to GL(n)$. The entry says something about spin structure, warning that the group $Spin(n)$ is not a subgroup of $GL(n)$. So what is meant ? The total space of a subbundle is a subspace at least. Does this mean that I consider the frame bundle first as a (non-principal) $Spin(n)$-bundle by pulling back along a fixed noninjective map $Spin(n)\to GL(n)$ and then I restrict to a chosen subspace on which the induced action of Spin group is principal ?

• I am working on entries related to generalized complex geometry. My aim is to tell the story in the correct way as the theory of the Lie 2-algebroid called the

Most, if not all, of the generalized complex geometry literature, uses the "naive" definition of Courant algebroids that regards them as vector bundles with some structure on them and is being vague to ignorant about what the right morphisms should be.

As discussed at Courant algebroid we know that we are really dealing with a Lie 2-algebroid and hence know where precisely this object lives. This provides some useful, I think, perspectives on some of the standard constructions. I want to eventually describe this. For the moment I have just the material at standard Courant algebroid with only two most basic observations (which, however, in my experience already take a nontrivial amount of time on the blackboard to explain to somebody used to the "naive" picture usually presented in the literature).

• brief category:people-entry for hyperlinking references at G-structure and at holographic QCD

(if anyone has a webpage for this author, let’s add the URL)

• Created Ho(Cat), mainly as a place to put a counterexample showing that it doesn't have pullbacks. If anyone has a simpler one, please contribute it.

• This is intended to continue the issues discussed in the Lafforgue thread!

I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.

I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.

Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.

• some trivial edits to opetope (a toc, some hyperlinks)

• Matthew Headrick, Lectures on entanglement entropy in field theory and holography (arXiv:1907.08126)
• a stub, for the moment just so as to satisfy links and record references

• I have split off from holographic principle and then expanded a good bit a few paragraphs on