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I have created a table
and included it into the relevant entries.
It looks like this:
| geometric context | universal additive bivariant (preserves split exact sequences ) | universal localizing bivariant (preserves all exact sequences) | universal additive invariant | universal localizing invariant |
|---|---|---|---|---|
| noncommutative algebraic geometry | noncommutative motives | noncommutative motives | algebraic K-theory | non-connective algebraic K-theory |
| noncommutative topology | KK-theory | E-theory | operator K-theory | … |
I am not sure what should go in the bottom right corner.
stub for non-connective algebraic K-theory
you all knew it was about to happen, and now it did: I have created a table-of-contents page for inclusion as “floating TOC” in the relevant entries:
added to limit in a quasi-category (in the Properties-section) more details on how the definition in terms of over over-categories is equivalent to the one in terms of homs. But still not done.
started orthogonal factorization system in an (∞,1)-category in order to record the n-connected/n-truncated system for .
Notice that for this application our “connected”-convention is really nicer than the “connective” convention.
added Soibelman’s lectures notes to the list of references at motivic Donaldson-Thomas invariant
quick note on motivic function
New articles: exponential map, logarithm.
added the definition of relative (infinity,1)-limit, because I need to point to it from elsewhere.
(Somehow the cache bug, or a new version of it kicked in, and there is now also relative (∞,1)-limit, which I won’t try to fight with just right now…)
(I’m currently reading through Keller’s book Differential Calculus in Locally Convex Spaces.)
bivariant algebraic K-theory – for the moment just for recording some references
I wrote partial differentiation. Also some remarks at chain rule.
Someone (probably in Gottingen) created a slideshow page. I renamed it Slideshow sandbox and removed its contents which were
www.math.columbia.edu/~woit/notes16.pdf
That was the old name as well.
stub for cosmic Galois group
briefly started numerical motive .
Urs created submersion and I added a little more. Still a bit stubby, though.
I have added the Nadler’s today’s paper at Waldhausen S-construction. It would perhaps make sense to quote it in some more entries.
Someone has made a change to the page on Aleksandr Aleksandrov, but there are some characters in the Cyrillic form of his name that are not coming out. Does anyone know how to fix this?
am starting to make some basic notes on Gysin/Umkehr/pushforward maps in KK-theory, currently in fiber integration in the section In KK-theory. But still with some gaps.
started stub for Chow group
hoping I got this right...
to idempotent completion (Karoubi envelope) I have added :
a pointer to further constructions discussed at Cauchy complete category
statement of the finality of the completion morphism;
a classical Reference
Started smooth loop space, initially just a stub. Partly to contain some bits of general theory relating to which smooth paths do I use (davidroberts) and partly to start transferring some notes on the differential topology of loop spaces over to the nLab.
In looking for somewhere to graft it on to the current nLab tree, I encountered loop space object. It seemed to me that the smooth loop space is not a loop space object, so I commented as such (thus also creating the link to smooth loop space which was my real intent). Someone who knows these things better than I do should check this.
created product law, since I wanted to be able to link to it…
Created unintentional type theory.
created an entry type II supergravity Lie 2-algebra.
started index theory - contents (somewhat optimistically, for the moment)
Created the page unbounded topos, and some links at topos and bounded geometric morphism.
I’m interested in the generalisation of the construction of the unbounded topos to the general case of an inaccessible comonad on a bounded topos (which wlog we might as well take to be EDIT: NO, LET’S NOT). In essence, why is it unbounded? Also, what nice properties can we claim of the category of coalgebras for , given information about .
Note also, the paper HOW LARGE ARE LEFT EXACT FUNCTORS? in TAC in 2001 seems to claim something a little stronger than Johnstone does in the Elephant, and recounted at topos, namely that the existence of lex endofunctors of set is independent of ZFC (they say something more general, but it covers this case). This is mostly a note to myself, but if others feel like looking, that would be good too.
You may have seen in “Recently revised” that I had edited 11-dimensional supergravity in the last days. I wanted to start a section there on the details of the action functional. But after adding some formulas, I ran out of time and just left an “under construction”-warning.
The reason I ran out of time is that I had to first write related things with higher priority into an article we are currently preparing:
Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory.
Later when the dust has settled and I have more leisure, I’ll try to take care of the Lab’s 11d sugra article again.
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.