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I have added to orthogonal factorization system
in the Definition-section three equivalent explicit formulations of the definition;
in the Properties-section the statement of the cancellability property.
Wanted to add more (and to add the proofs). But have to quit now. Maybe later.
I treid to clean up Whitehead tower a bit:
I rewrote and expanded the Idea/Definition part.
Then I moved David Roberts' material that was there to the appropriate section at the new Whitehead tower in an (infinity,1)-topos. (There I tried to add some introductory remarks to it but will try to further highlight David's results here in a moment).
At Whitehead tower I left just a new section that says that there is a notion of Whitehead towers in more general contexts with a pointer to Whitehead tower in an (infinity,1)-topos
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.)
I finally gave the Connes-Lott-Chamseddine-Barrett model its own entry. So far it contains just a minimum of an Idea-section and a minimum of references.
This was prompted by an exposition on PhysicsForums Insights that I wrote: Spectral standard model and String compactifications
tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.
as mentioned in another thread, I have expanded the Idea-section at polarization in order to highlight the relation to canonical momenta (which I also edited accordingly).
started an Examples-section at geometric quantization
there is an old article (Berends-Gastman 75) that computes the 1-loop corrections due to perturbative quantum gravity to the anomalous magnetic moment of the electron and the muon. The result turns out to be independent of the choice of (“re”-)normalization (hence what they call “finite”).
I have added a remark on this in the (g−2)-entry here and also at quantum gravity here.
I have been expanding and polishing the entry Heisenberg group.
This had existed in bad shape for quite a while, but now it’s maybe getting into better shape.
I tried to spend some sentences on issues which I find are rarely highlighted appropriately in the literature. So there is discussion now of the fact that
there are different Lie groups for a given Heisenberg Lie algebra,
and the appearance of an “i” in [q,p]=i may be all understood as not picking the simply conncted ones of these;
I also added remarks on the relation to Poisson brackets, and symplectomorphisms.
In this context: either I am dreaming, or there is a mistake in the Wikipedia entry Poisson bracket - Lie algebra.
There it says that the Poisson bracket is the Lie algebra of the group of symplectomorphisms. But instead, it is the Lie algebra of a central extension of the group of Hamiltonian symplectomorphisms.
I took the liberty of incorporating material from Andre Joyal's latest message to the CatTheory mailing list into the entry dagger-category:
created sections
added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors
Initial stub to record some references. Wanted by type theoretic model category
The entry test category which I wrote some time ago, came into the attention of Georges Maltsiniotis who kindly wrote me an email with a kind praise on nlab and noting that his Astérisque treatise on the topic of Grothendieck’s homotopy theory is available online on his web page and that the Cisinski’s volume is sort of a continuation of his Astérisque 301. Georges also suggested that we should emphasise that a big part of the Pursuing Stacks is devoted to the usage of test categories, so I included it into the bibliography and introductory sentence. I hinted to Georges that when unhappy with a state of an nlab entry he could just feel free to edit directly.
Have added to pushout-product the statement (here) that pushout product □ of I1-cofibrations with I2-cofibrations lands in (I1□I2)-cofibrations; and (here) the example of pushout products of the inclusions Sn−1↪Dn. Both without proof for the moment.
Added a paper:
I fixed a link to a pdf file that was giving a general page, and not the file!
the book Monoidal Functors, Species and Hopf Algebras is very good, but still being written. Clearly the current link under which it is found on the web is not going to be the permanent link. So I thought it is a bad idea to link to it directly. Instead I created that page now which we can reference then from nLab entries. When the pdf link changes, we only need to adapt it at that single page.
created nilradical
I felt that we had too many gray links to metalanguage, so I gave it a try. But I don’t really have the leisure for it now and not the expertise anyway. Experts please feel invited to take apart what I wrote there and replace with it something better.
added an Idea-section to Mackey functor (which used to be just a list of references). Also added more references.
In the past we had some discussion here about why simplicial methods find so much more attention than cubical methods in higher category theory. The reply (as far as I am concerned at least) has been: because the homotopy theory = weak oo-groupoid theory happens to be well developed for simplicial sets and not so well developed for cubical sets. Historically this apparently goes back to the disappointment that the standard cubical geometric realization to Top does not behave as nicely as the one on simplicial sets does.
Still, it should be useful to have as much cubical homotopy theory around as possible. Many structures are more naturally cubical than simplicial.
So as soon as the Lab comes up again (we are working on it...) I want to create a page model structure on cubical sets and record for instance this reference here:
Jardine, Cubical homotopy theory: a beginning
added pointer to:
a bare list of references, to be !include-ed into the References-sections of relevant entries (such as abelian Chern-Simons theory and fractional quantum Hall effect) for ease of synchronization
Made some further tweaks at cubical set. Hopefully the definitions of the boundary functor, and of a horn, are correct now.
Am continuing to work on homotopy groups of a cubical Kan complex.
Orphaning this page after merging into category of cubes.
Hello,
I noticed DFT page has not been updated in a while and I added a couple of sections: some sketchy introductory material (analogy between Kaluza-Klein and DFT) and a little insight about a more rigorous geometrical formulation of DFT.
It is still quite sketchy but I would be happy to refine it.
PS: this is my first edit, I hope I played by the rules. And thank you all for this wiki
Luigi
added brief pointer to the derivation of SO(32) gauge group via tadpole cancellation, and some references on type I phenomenology. Will add these also to string phenomenology and to GUT, as far as relevant there
Created:
Every Stein manifold of dimension n admits an injective proper holomorphic immersion into C2n+1.
Every holomorphically complete complex space of dimension n admits an injective proper holomorphic map into C2n+1 that is an immersion at every uniformizable point.
If for some N>n a holomorphically complete complex space X is locally isomorphic to an analytic subset of an open set in CN, then there is an injective proper holomorphic map ϕ:X→CN+n that is an isomorphism onto its image.
The relevant spaces of embeddings are dense in the space of all holomorphic mappings into the corresponding cartesian spaces equipped with the compact convergence topology.
The original reference is
Created:
Every real-analytic n-dimensional second countable Hausdorff manifold admits a real-analytic, regular and proper embedding into a euclidean space Rk of sufficiently high dimension.
The original reference is Theorem 3 in
Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.
Started literature section with several references at forcing.