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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.
I began to add a definition of conformal field theory using the Wightman resp. Osterwalder-Schrader axiomatic approach. My intention is to define and explain the most common concepts that appear again and again in the physics literature, but are rarely defined, like “primary field” or “operator product expansion”.
(I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem, I don’t remember reading or hearing an answer of that until I looked in the book by Schottenloher).
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 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 admits an injective proper holomorphic immersion into .
Every holomorphically complete complex space of dimension admits an injective proper holomorphic map into that is an immersion at every uniformizable point.
If for some a holomorphically complete complex space is locally isomorphic to an analytic subset of an open set in , then there is an injective proper holomorphic map 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 -dimensional second countable Hausdorff manifold admits a real-analytic, regular and proper embedding into a euclidean space 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.
I am at the Croatian black hole school organized by Jarah Evslin, and I am partially taking care of Croatia related issues (visa, trasnportation advice. communication to the owners of the housing). Lots of interesting things here about star formation, black hole formation, making massive black holes from lighter ones and so on. And some string theory mechanisms related to black hole entropy and similar issues. Most of people are postdocs and students here. Among seniors, Holger Nielsen and Mina Aganagić are present to our benefit.
By the way, started a stub black hole. Please contribute.
started something. For the moment really just a glorified pointer to Buchert et al. 15 and putting Scharf 13 into perspective
am finally splitting this off from Hopf invariant
I have added at HomePage in the section Discussion a new sentence with a new link:
If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.
I had completly forgotton about that page Welcome to the nForum (nlabmeta). I re-doscivered it only after my recent related comment here.
Created:
Just like an ordinary scheme in algebraic geometry is glued from affine schemes, a C^∞-scheme in differential geometry is glued from smooth loci.
The original reference is
See the artice C^∞-ring for more references.
starting something.
I claim that in terms of quantum circuits via dependent linear types, the principle of deferred measurement is immediately formalized and proven by the Kleisli equivalence:
Namely a quantum circuit involving measurement in the -basis anywhere is a Kleisli morphism for the linear necessity-comonad, and the Kleisli equivalence says that this equals a coherent (non-measurement) quantum circuit postcomposed with the -counit: But the latter is the measurement gate.
Change 1: Original page describes the fan theorem as requiring the bar to be decidable, claims that the “classical” fan theorem contradicts Brouwer’s continuity principle. The latter claim is not true; I corrected the error. I have stated the result as two separate theorems: the decidable fan theorem, about decidable bars, and the fan theorem, about bars in general.
Change 2: Slightly more information is provided about the relationship between the Fan Theorem and Bar Induction. Eventually, we should make a page about the latter.
Change 3: the section on equivalents to the fan theorem has been fixed somewhat. The section originally asserted that all of the statements provided were equivalent to the decidable fan theorem; in fact, some are equivalent to the decidable fan theorem and some to the full fan theorem.
moving material about the lesser limited principle of omniscience from principle of omniscience to its own page at lesser limited principle of omniscience
Anonymouse
After a suggestion from Toby, I added a note on the “analytic Markov’s principle” to Markov’s principle.
moving material about the weak limited principle of omniscience from principle of omniscience to its own page at weak limited principle of omniscience
Anonymouse
moving material about the limited principle of omniscience from principle of omniscience to its own page at limited principle of omniscience
Anonymouse
Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.
Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.
I would try to say what the idea is, but have to dash.
added pointer to yesterday’s
added doi-link to
this is a bare subsection with a list of references, meant to be !include-ed into the References-lists of relevant entries (such as AdS-QCD correspondence, AdS-QCD correspondence but also at flux tube and maybe at string)
this MO comment made me realize that we didn’t have an entry proof assistant, so I started one
added some references on constructive analysis here.
I have expanded a bit at Serre-Swan theorem: gave it an actual Idea-section, mentioned more variants (over general ringed spaces, in higher geometry) and added more references.
New article class equation, just to fill some gaps in the nLab literature. Truly elementary stuff.