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added to hypercomplete (infinity,1)-topos a comment on how classical topos theory models these
(motivated from our discussion here)
wrote something at cosmic string and by copy-and-pasting-and-changing-the-degrees added something similar to domain wall and monopole. Needs a bit more polishing, maybe.
I am really working on writing out an abstract re-formulation of this classical theory in terms of extended TQFT with defects, but not done with that yet (and will probably be interrupted again before finishing it).
Added pointers to
to relevant entries, such as Dijkgraaf-Witten theory and Topological Quantum Field Theories from Compact Lie Groups.
There used to be a warning at infinity-Lie algebra cohomology about whether or not a certain functor needs to be regarded as a derived functor in order to get the correct homotopy-theoretic interpretation of oo-Lie algebroid extensions. I think I have now spelled out at synthetic differential infinity-groupoid the required details and so I replaced that warning with a pointer to a section in that latter entry.
All this can do with a good bit more polishing. I’ll see what I can do eventually.
created symplectic connection, just for completeness at deformation quantization
I added a link to EoM at Lie’s three theorems, where there is a statement of the first theorem - we just had the slightly disparaging sentence
…is today regarded as lacking a good notion of differentiable manifold.
stub for Tamagawa number, for the moment just so as to record the article relating it to YM theory.
I archive here the query box from homological algebra and finite element method.
Eric: The appearance of homological algebra in finite element methods goes back a lot earlier than Arnold. It is covered somewhat extensively in Robert Kotiuga’s (who happened to be John Baez’s dorm room mate at MIT) PhD dissertation:
- Kotiuga, P.R., Hodge Decompositions and Computational Electromagnetics, Ph.D. Thesis, McGill University, Montreal, Canada, 1985
I don’t think Robert would claim to be the earliest.
Then you can go back to 1976 with Józef Dodziuk’s classic paper:
- Finite difference approach to the Hodge theory of harmonic forms, Amer. J. Math., 98 (1976), 79-104.
Zoran: Surely, it is emphasised that the numerous precursors exist in the Bulletin survey of Douglas Arnold et al. quoted below. But it was not in any sense systematic till rather recently. One could expand on the history…
I have today posted a question to MathOverflow on recent-fundamental-new-directions-in-pdes.
Anon. has created an empty maximal geodesic.
The old entry 1-groupoid was a bit vague. I have added a paragraph with a more precise description.
Note: Anon… has created discrete logarithm but has not entered any content.
I started convex cone. While imitating the link from Jordan algebra to cone, shouldn’t there be a disambiguation between the category theoretic cone and the vector space cone? But even as it is, cone is not that clear on the relation between the cone of homotopy theory and the cone of category theory.
I am starting a table
noncommutative geometry - contents
to be included as a “floating table of contents” in the relevant entries.
Clearly this is just a beginning. Zoran will have lots of items to add.
Look at what has been happening at derivator. The entry was erased and various things put there by an Anonymous Coward. Another one has reinstated the original one!! Has anyone noticed this? I was travelling about the time it happened so my usual check did not occur.
At triangle identities something similar had started. I have rolled back.
A stub for Cartan-Eilenberg categories.
created a table homotopy n-type - table and included it into the relevant entries
While creating double Lie algebroid I notied that we had a neglected entry double groupoid. I gave it a few more lines.
Wallman compactification, redirecting also Wallman base (previously wanted at Stone Spaces). See the link to videos by Caramello, where also Alain Connes participates in a discussion.
I felt we needed an entry explicitly titled noncommutative stable homotopy theory. So I created one. But it’s just a glorified redirect to KK-theory and E-theory.
added references to symplectic leaf
New stubs scattering and abstract scattering theory.
Chevalley’s theorem on constructible sets and elimination of quantifiers. The entries are related ! The interest came partly from teaching some classical algebraic geometry these days. The related entry is also forking, though yet it is not said why; non-forking may be viewed as related to a notion of generic point, generic type (in the sense of model theory).
Was this meant to be Planck Collaboration? I have renamed it!
To add the old entry birational geometry I added a number of classical, very geometric, algebraic geometry entries rational map, birational map, rational variety, image of a rational map, unirational variety and a number of redirects. The notion of an image is a bit unusual because the varieties and rational maps do not make a category, as the composition is not always defined. However the notion of the image is still very natural here. For the concept of dominant rational map I did not make a separate entry but discussed it within rational map and made redirects.
I moved much of material from Hopf algebroid to Hopf algebroid over a commutative base, where both groupoid convolution algebras and group function algebras belong there. Most of the difference is seen already at the level of bialgebroid, the stuff about antipode in general case is to be written. Some more changes to both entries.
A stub for metric abstract elementary class. Related changes/additions on some model theory entries like elementary class of structures, forking entered a related blog at math blogs.
created cofinal (infinity,1)-functor
added to slice (infinity,1)-category the statement that projecting slicing object away (dependent sum) reflects ∞-colomits.
For some reason, we never had map redirect to function, but now we do. Same with mapping.
This may or may not be the best behaviour. We might actually want a page on how people distinguish these words, such as in topology (‘map’ = continuous map but ‘function’ = function, maybe).
started field (physics).
So far there is an Idea-section, a general definition with some remarks, and the beginning of a list of examples, which after the first spelled out (gravity) becomes just a list of keywords for the moment.
More later.
Added to A-infinity category the references pointed to by Bruno Valette here.
Created binary Golay code. The construction is a little involved, and I haven’t put it in yet, because I think I can nut out a nicer description. The construction I aim to describe, in slightly different notation and terminology is in
R. T. Curtis (1976). A new combinatorial approach to M24. Mathematical Proceedings of the Cambridge Philosophical Society, 79, pp 25-42. doi:10.1017/S0305004100052075.
I noticed that we have kinematic tangent bundle.
To incorporate this a bit into the nLab -web I have created stubs for operational tangent bundle (wanted by its kinematic cousin) and for synthetic tangent bundle and then I have interlinked all these entries and linked to them from tangent bundle.
Also gave the Idea-section of kinematic tangent bundle a very first paragraph which very briefly says it all, before diving into discussion of what generalized smooth spaces are etc.
created homotopical structure on C*-algebras , summarized some central statements from Uuye’s article on structure of categories of fibrant objects on C*Alg.
added to some relevant entries a pointer to
There is a new stub E-theory with redirect asymptotic morphism, new entry semiprojective morphism (of separable C*-algebras) and stub Brown–Douglas–Fillmore theory, together with some recent bibliography&links changes at Marius Dadarlat, shape theory etc. There should be soon a separate entry shape theory for operator algebras but I still did not do it.
created entries trialgebra and Hopf monoidal category
also expanded the Tannaka-duality overview table (being included in related entries):
to contain the first entries of the corresponding “higher Tannaka duality” relations
added to Hilbert bimodule a pointer to the Buss-Zhu-Meyer article on their tensor products and induced 2-category structure.
I have now created relative category.
Question: Does the transferred model structure on RelCat resolve Rezk’s [2001] conjecture that the classification diagram of a model category is weakly equivalent to its simplicial localisation? The Nξ functor looks very close to computing the hammock localisation to me…
cross-linked weak Hopf algebra and fusion category a bit more explicitly, added to both a reference to Ostrik’s article that shows the duality and added a corresponding item to
I have added to groupoid convolution algebra the beginning of an Examples-section titled Higher groupoid convolution algebras and n-vector spaces/n-modules.
Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a “higher C*-algebra” version of convolution algebras of higher Lie groupoids.
I have labelled it as “under construction” to reflect the fact that this latter aspect is a bit experimental for the moment.
The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor
C:Grpd→Algopb≃2Mod(as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends double groupoids to convolution sesquialgebras, hence to 3-modules with basis (3-vector spaces).
As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the “horizontally constant” double groupoid incarnation of BG, while the convolution algebra of G is the algebra of the “vertically discrete” double groupoid incarnation of BG.
But next, if we simply replace the bare Algopb≃2Mod with the 2-category C*Algb of C*-algebras and Hilbert bimodules between them and assume (as seems to be the case) that C*-algebraic groupoid convolution is a 2-functor
LieGrpd≃→C*Algopnthen the same argument goes through as before and yields convolution “C*-2-algebras” that look like Hopf-C*-algebras. Etc. Seems to go in the right direction…
seeing the announcement of that diffiety summer school made me think that we should have a dedicated entry titled cohomological integration which points to the aspects of this discussed already elswhere on the nLab, and which eventually lists dedicated references, if any. So I created a stub.
Does anyone know if there is a published reference to go with the relevant diffiety-school page ?
dropped some lines into a new Properties-section in the old and neglected entry bibundle. But not for public consumption yet.
I felt we were lacking an entry closure operator. I have started one, but don’t have more time now. It’s left in a somewhat sad incomplete state for the moment.
just noticed that this morning some apparently knowledgable person signing as “Snoyle” added two paragraphs to finite group with technical details.
I have helped a bit with the syntax now and split off entries for quasisimple group and generalized Fitting subgroup
started Hopf C-star algebra (but my computer is running out of battery power now..)
brief note: canonical Hilbert space of half-densities
Added brief comments at star algebra, at dagger category and at category algebra on how convolution algebras on dagger-categories are naturally star algebras.
I made a stub Tannakian category with some references.
Added the recent reference on Langlands dual groups as T-dual groups to both geometric Langlands correspondence and T-duality together with a brief sentence. But nothing more as of yet.
I could have sworn that we already had entries like “topological ring”, “topological algebra” or the like. But maybe we don’t, or maybe I am looking for the wrong variant titles.
I ended up creating a stub for topological algebra now…
I have added to C-star algebra the statement that the image of a C*-algebra under an *-homomorphism is again C*.
Also reorganized the Properties-section a bit and added more references.
the entry groupoid could do with some beautifying.
I have added the following introductory reference:
I have started adding some references to
on modules (C*-modules) of (continuous, etc..) convolution algebras of topological/Lie groupoids.
I still need to look into this more closely. A motivating question for this kind of thing is:
what’s the right fine-tuning of the definition of modules over twisted Lie groupoid convolution algebras such that for centrally extended Lie groupoids it becomes equivalent to the corresponding gerbe modules?
This seems fairly straightforward, but there are is some technical fine-tuning to deal with. I was hoping this is already stated cleanly in the literature somewhere. But maybe it is not. Or maybe I just haven’t seen it yet.
Wrote a quick note at centrally extended groupoid and interlinked a little, for the moment just motivated by having the link point somwhere.