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I was reading Adams’ lectures on generalised cohomology theories and added some stuff from there to universal coefficient theorem about the more general case (including the Kunneth theorem).
Another new article: sequence space. I await the inevitable report that this term is also used for other things.
New page: Banach coalgebra.
Hopefully you all know that is a Banach algebra under convolution, but did you know that is a Banach coalgebra under nvolution? (Actually, they are both Banach bialgebras!)
I added an Idea-section to element in an abelian category and added a reference by George Bergman.
This links back to the new Idea section at abelian categories - embedding theorems. Check if you agree with the wording.
I have added to the entry chain homology and cohomology an actual Idea-section and an actual definition. The material that used to be there I have moved into a section Chain homology – In the context of homotopy theory.
stub for Thomspon’s group F.
created a little table: chains and cochains - table and included it into the relevant entries (some of which still deserve to be edited quite a bit).
discussion over in the thread on modular lattice keeps making me create stub likes unimodular lattice and Leech lattice. No genuine content there yet, though. It just seems necessary to have these entries at all.
I am splitting off Heisenberg Lie n-algebra from Heisenberg Lie algebra .
I have created a table relations - contents and added it as a floatic TOC to the relevant entries.
statement of the snake lemma
Is there a reason for the page cohomology theory to exist independently, rather than as a redirect to generalized (Eilenberg-Steenrod) cohomology?
Created delta-functor.
(also touched Tohoku, adding hyperlinks and “the”-s)
I added a few observations under a new section “Results” at bornological set. Bornological sets form a quasitopos. I don’t have a good reference for the theorem of Schanuel.
Related is an observation which hadn’t occurred to me before: the category of sets equipped with a reflexive symmetric relation is a quasitopos. I’d like to return to this sometime in the context of thinking about morphisms of (simple) graphs.
I have started an entry (∞,n)-category with adjoints, prompted by wanting to record these slides:
If anyone can say more about the result indicated there, I’d be most grateful for a comment.
Also, I seem to hear that at Luminy 2012 there was some extra talk, not appearing on the schedule (maybe by Nick Rozenblyum, but I am not sure) on something related to geometric quantization. If anyone has anything on that, I’d also be most grateful.
New page dual space with redirect from dual basis.
I am splitting off 2-plectic geometry from n-plectic geometry .
I am starting a table of contents, to be included as a floating TOC for entries related to duality:
But it’s a bit rough for the time being. I haven’t decided yet how to best organize it and I am probably still lacking many items that deserve to be included. To be developed. All input is welcome.
Created product-preserving functor.
New article: direct sum of Banach spaces. These come in even more variety than I originally thought!
I got tired of writing ‘short linear map’, so now we have short linear map.
I created function application, so as to be able to link to it from fixed-point combinator. While adding links, I was motivated to expand a bit on function.
I started the article Z-infinity-module. Hopefully someone here can say something more interesting about them!
I added some remarks to adjoint equivalence about improving equivalences to adjoint ones, with links to&from equivalence and equivalence in a quasicategory.
I'm putting all the big duality theorems from measure theory at Riesz representation theorem. Only a couple are filled in so far, but I'm out of time for today.
There have been several entries with no math content recently. What is the procedure? I am meaning: Search results for definion and The Enemy of my Enemy is not my Friend.
Heya. I haven’t actually made the necessary changes, but the various pages on dependent type theory make the statement that every DTT or MLTT is the internal logic of an LCCC and every LCCC is the categorical semantics of some DTT/MLTT. However, this is extremely confusing (it took me 2 or 3 hours to find a page where it was made completely clear), since it makes explicit use of super-strong extensionality (I think this is called beta-translation), that is to say, it is a theorem about extensional DTTs/MLTTs.
It’s not even totally clear to me that every intensional type theory actually has an (∞,1)-categorical semantics without the consideration of the univalence axiom. I would make this clearer, but I am really out of my depth with type theories, so I’m just alerting you to the fact that this is stated confusingly almost everywhere (the only place where it’s clear is in the page on identity types).
Not much here, but: predual.
New page: inclusion-exclusion principle
New page: positive cone, including the extended positive cone of a W*-module.
Wrote Lambert W function. It was an excuse to record Joyal’s proof of Cayley’s theorem on the number of tree structures one can put on an -element set (which is ).
I’ve been inactive here for some months now; I hope this will significantly change soon.
I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:
Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?
Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?
In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products , and form a new 2-category with 2-products whose 0-cells are pseudomonoids in , whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate , starting with . Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace with (where all the units are forced to coincide up to isomorphism)?
Apologies if these are naive questions; I am not very familiar with the literature.
I created a stub on excision, but this is just a link to the Wikipedia page for the moment.
Concrete, abstract: group actions, groups; concrete categories, categories; Cartesian spaces, vector spaces; von Neumann algebras, -alebras; material sets, structural sets; etc. At concrete structure.
I wrote an overview over some constructions on- and examples of group schemes.
as some of you will have seen, I had spent part of the last week with attending talks at String-Math 2012 and posting some notes about these, to the Café (here). For many of these notes I added material to existing Lab entries (mostly just references) or created Lab entries (mostly just stubs).
But since at the same time I was also finalizing the writup of an article as well as doing yet some other things, the whole undertaking was a bit time-pressured. As a result, I decided it would be too much to announce every single Lab edit that I did here on the Forum.
So I ask you for understaning that hereby I just collectively announce these edits here: those who care should please scan through the list of blue links here and see if they spot pointers to Lab entries where they would like to check out the recent edits.
I think I can guarantee, though, that in all cases I did edits that should be entirely uncontroversial, their main defect being that in many cases they leave one wish for more exhaustive discussion.
added some references by Catahrina Stroppel at the end of categorification in representation theory
(also added the words “representation theory” to the entry itself :-)
I've been meaning to write this for a while. Now I need to look at Bourbaki this weekend to explain their approach.
New page: Radon–Nikodym derivative.
I moved some material from state to create pure state (redirect from mixed state).
I have created a stub quantum affine algebra as a means to collect some references, alluded to here.
If there is any expert on the matter around, he or she should please feel invited to add an illuminating Idea-section to the entry.
I created types and calculus and seven trees in one. Both entries as yet contain just references.
It would be nice to have more articles expanding on the reltion of calculus and (higher) category theory /type theory.
created conjugacy class
New stub metaplectic representation, for now containing only some references.
Maybe I am not searching correctly, but it seems to me that until 2 minutes ago the rather remarkable diagram of LCTVS properties was linked to from exactly none non-meta Lab page. It was effectively invisble unless one explicitly searched for “SVG”.
Let me know if there is a reason for it remaining invisible. Assuming that there isn’t, I have now added it to locally convex space and to functional analysis - contents (which I restructured slightly, moving the two such overview diagrams prominently to the top, where they can be recognized as what they are).
stub for metaplectic structure
added a few more Examples to reduction of structure groups.
created metalinear structure. Added it to square roots of line bundles - table . Linked to it from Theta characteristic and so forth.
started metalinear group
Danny Stevenson was so kind and completed spelling out the proof of the pasting law for -pullbacks here at (infinity,1)-pullback.
I created a stub for Kirchhoff’s laws to go with the Café-discussion here. Maybe somebody feels like expanding it, I don’t really have the time for this right now.
I have created an entry symplectic reduction - table and included it into relevant entries.
I wrote Hamiltonian action.
I tried to say precisely what the action is by. In the literature (but also in a previous version of our moment map entry) there is often (for instance on Wikipedia, but also in many other sources) an imprecise (not to say: wrong) statement, where an action by Hamiltonian vector fields is not distinguished from one by Hamiltonians.
I have decided to splitt off a stand-alone entry symplectic reduction from BRST-BV formalism (which used to be the redirect). Still just a stub. Lots of material and references still needs to be copied or moved from the latter to the former.