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I’ve added the below to the Idea section of action#idea as a simpler intro before jumping into delooping. Maybe some of the text in the footnote should be incorporated into the body, and I haven’t changed anything that follows to jibe with it.
The simplest notion of action involves one set, X, acting on another Y as a the function act:X×Y→Y. This can be curried as ^act:Y→YX where YX is the (monoidal) set of functions from X to Y.1
Generalized notions of action use entities from categories other than Set and involve an exponential object such as YX.
In the category Set there is no difference between the above left action and the right action actR:Y×X→Y because the product commutes. However for the action of a monoid on a set (sometimes called M-set or M-act) the product of a monoid and a set does not commute so the left and right actions are different. The action of a set on a set is the same as an arrow labeled directed graph arrows:vertices×labels→vertices which specifies that each vertex must have a set of arrows leaving it with one arrow per label, and is also the same as a simple (non halting) deterministic automaton transition:inputs×states→states. ↩
Grothendieck-Teichmüller tower with redirect Grothendieck-Teichmüller group.
added to frame bundle the definition of its canonical 1-form
That’s not well put at flag variety, is it?
More generally, the generalized flag variety is the complex projective variety obtained as the coset space G/T≅Gℂ/B where G is a compact Lie group, T its maximal torus, Gℂ the complexification of G, which is a complex semisimple group, and B⊂Gℂ is the Borel subgroup. It has a structure of a compact Kähler manifold. It is a special case of the larger family of coset spaces of semisimple groups modulo parabolics which includes, for example, Grassmannians.
The ’larger family’ are the generalized flag varieties, no?
I created a stub for stabilizer subgroup as it was needed by orbit.
This could do with a nPOV section, but I am not sure what to say for that. (I am still not sure how to ‘do’ an nPOV version of all the stuff around group actions. I would love to have it clear in my head as then a straightforward nPOV adaption of Grothendieck in SGA1 would be feasible. The treatment of SGA1 in the nLab still has some holes in it… e.g. the transition from the prorepresenting P to the profinite group Aut(P). I hope to sketch a bit more detail there soon. In fact a lot of the detail is skated over in the original SGA1. Any thoughts anyone?)
Someone (anonymous) has raised a query at quality type. They say:
(Is “consists entirely of idempotents” correct?)
Added pointers to Selberg zeta function for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function exp(iπηD(0)) equals the Selberg zeta function of odd type.
Together with the fact at eta invariant – For manifolds with boundaries this says that the Selberg zeta function of odd type constitutes something like an Atiyah-style TQFT which assigns determinant lines to surfaces and Selberg zeta functions to 3-manifolds.
This brings me back to that notorious issue of whether to think of arithmetic curves as “really” being 2-dimensional or “really” being 3-dimensional: what is actually more like a Dedekind zeta function: the Selberg zeta functions of even type or those of odd type?
I have started some bare minimum in entries
and cross-linked a bit.
created a bare minimum at Diophantine equation, just for completeness.
Also made Diophantine geometry a redirect to arithmetic geometry and added there one line saying way.
added to the people-entry Maxim Kontsevich in the list of the four topics for which he received the Fields medal brief hyperlinked commented on what these keywords refer to.
created etale geometric morphism
(david R.: can we count this as a belated reply to your recent question, which I can’t find anymore?)
I am writing some notes for a talk that I will give tomorrow:
I thought this might serve also as an exposition for a certain topic cluster of nLab entries, so I ended up typing it right into the nLab.
Notice that this is presently a super-rough version. At the moment this is mostly just personal jotted notes for myself. There will be an abundance of typos at the moment and at several points there are still certain jumps that in a more polished entry would be expanded on with more text.
So don’t look at this just yet if you have energy only for passive reading.
This is failing to load (on Firefox). I have tried several times and the error message changes each time! The latest was:
XML Parsing Error: no element found Location: http://ncatlab.org/nlab/show/category+of+fibrant+objects Line Number 1350, Column 39: (N \mathbf{B}(-,G))_1 : U \mapsto C( ————————————–^
I have also tried it in Safari and the page does not load beyond a certain point.
I think there was some terminological confusion, where the nLab defined Segal’s category Γ to be a skeleton of finite pointed sets; I think it should be the category opposite to that. I’ve made edits at this article and at Gamma-space.
added a paragraph about passing from first-order logic to modal type theory to the entry on analytic philosophy.
Stub for double-negation topology.
An equivariant derived category is **not* an example of a derived category of an abelian category in general, it has a more intricate construction. In the case of the free action it is equivalent to the derived category of the abelian category of equivariant sheaves. I have created just an idea section and wrote down the main references. The apporpriate treatment has been discovered by Valery Lunts and Joseph Bernstein. I just created the entry for the latter. Changes/links/additions/redirects at Alexander Beilinson, equivariant sheaf.
Via, Math Stackexchange the location of Barr’s English translation of Grothendieck’s “Tohoku” has moved to http://www.math.mcgill.ca/barr/papers/gk.pdf. I’ve also updated the link on the nLab’s Tohoku page.
I’ve made comments at Fivebrane, fivebrane 6-group, Fivebrane structure and differential fivebrane structure regarding the fact String(n) is not 6-connected for n≤6 (though trivially so for n=2).
There will be a bunch of interesting invariants for manifolds of dimensions 3-6. This includes, for instance, on 6-dimensional spin manifolds a non-torsion class corresponding to the obstruction to lifting the tangent bundle to the 6-connected group covering String(6) (here π5(String(6))=ℤ). This is the only non-torsion example, but should be given by a U(1)-4-gerbe, I think, which will have a 6-form curvature. Since H6 won’t be vanishing for oriented manifolds, one has some checking to do. For instance, the frame bundle of S6 lifted to a String(6)-bundle (I plan to write a paper on this 2-bundle) will not lift to a Fivebrane(6)-bundle, because it won’t even lift to a ˜String(6)-bundle (i.e. the 6-connected cover of String(6)), since the transition function S5→BString(6) is the generator. Thus the 6-form curvature of this 4-gerbe should be the volume form on the 6-sphere.
Another point that occurs to me is that there are two copies of ℤ to kill off in String(8) to get Fivebrane(8), so one gets a U(1)×U(1) higher gerbe. I suspect this larger π7 is why there are so many more exotic spheres in dimension 15 than in neighbouring dimensions (16256 vs 2 in d=14 and 16); it’s certainly the case for exotic spheres in dimension 7 that π3(SO(4))=ℤ×ℤ helps.
Here we are going to explain the application of Topology.
have highlighted a bit more the fact here that the atoms in a subtopos lattice are the 2-valued Boolean ones. Thanks to Thomas Holder for alerting me.
And have added this as an example/proposition to atom and to Boolean topos, too.
created a brief entry K-theory of a symmetric monoidal (∞,1)-category.
In the course of this I have also split off a brief entry ∞-group completion from Grothendieck group and did some other cross-linking.
(The collection of entries on algebraic K-theory and its variants that we have would deserve a serious clean-up….)
I threw in some references to the early topos approach to set theory in ETCS. On this occasion I couldn’ t resist the temptation to rearrange somewhat the lay-out of the entry: actually I thought it better not to throw HOTT immediately at the reader and gave Palmgren’s ideas a proper subsection. I’ve also deflated a bit the foundational claims of ETCS sticking more to what appears to me to be Lawvere’s original intentions.
created essentially algebraic (infinity,1)-theory (just for completness, nothing unexpected there)
By the way, at essentially algebraic theory there is a pointer to “references below”, but no references are given.
started a hyperlinked index for the book Spin geometry
put a bare minimum into graded commutator, just because I needed to link to it.
wrote a bare minimum into Atiyah-Bott-Shapiro isomorphism.
<p>I created <a href="https://ncatlab.org/nlab/show/synthetic+differential+geometry+applied+to+algebraic+geometry">synthetic differential geometry applied to algebraic geometry</a> which is supposed to host a question that I am going to post on <a href="http://go2.wordpress.com/?id=725X1342&site=sbseminar.wordpress.com&url=http%3A%2F%2Fmathoverflow.net%2F">math Overflow</a> following the discussion we have of that <a href="http://sbseminar.wordpress.com/2009/10/14/math-overflow/#comment-6875">here at SBS</a>.</p>
<p>In that context I also wrote a section at <a href="https://ncatlab.org/nlab/show/algebraic+geometry">algebraic geometry</a> intended to describe the general-nonsense perspective. But that didn't quite find the agreement with Zoran and while we are having some discussion about this in private, he has restructured that entry now.</p>
started Calabi-Yau object. But am being interrupted…
brief entry on Turaev-Viro model, an entry long overdue. But for the moment it just records some references.
Bruce Bartlett has a comment on what is currently the last of these references and he will post it below in a moment…
am creating
listing the statements of the classification results, for the various cases. As far as I am aware of them.
Created a stub at Milnor K-theory, which is now just an MO answer of Cisinski. To be expanded at some later point when I study this in more detail.
New page: n-types cover
wrote the first part of a discussion of prequantized Lagrangian correspondences, showing how traditional Hamiltonian and Lagrangian mechanics are naturally absorbed into the context of “local prequantum field theory” and “motivic quantization”.
Simple as it is, but does anyone know if the proposition in the section The classical action functional prequantizes Hamiltonian correspondences has been made explicit in the literature before? I can’t find it, but it should have been discussed before. If anyone has a citation, please let me know. Of course all the ingredients of the little proof are simple classical steps, but I am wondering if this has been observed as a statement, simple as it may be, on the prequantum lift of the more famous Lagrangian correspondences.
I added the redirect anomaly to quantum anomaly to give a target to a link Zoran put on the Cafe. If this is counter to your intention for the link, Zoran, I can remove the redirect and put in a stub instead.
I added two new important references on global analytic geometry, also due to Poineau. He shows there that the sheaf of analytic functions is coherent. This is an interesting fundamental result.
started derived loop space
technical details to follow
Couldn’t find an existing “latest changes” thread for the quasifibration page (http://ncatlab.org/nlab/show/quasifibration), but just wanted to remark that I put another reference in there, to a nice expository paper by Peter May.
-Jon
Created lambda calculus.
I have added pointer to Mike’s discussion of spectral sequences here at “homotopy spectral sequence” and in related entries.
But now looking at this I am unsure what the claim is: has this been formalized in HoTT?
(Clearly a question for Mike! :-)
expanded the entry cofinal functor: formal definition, list of equivalent characterizations and textbook reference.
Just an editorial matter:
noticed that section did not point to space of sections and not to dependent product in the text. So I briefly added a paragraph to this extent.
How would you define the usual jargon “fragment” in logic?
There ought to be a simple formal definition, I suppose, such as “Given a language L and a theory T in that language, then a fragment of T is… “
created topological localization
added to periodic ring spectrum and to periodic cohomology theory a brief paragraph on looping/delooping periodicity on the ∞-modules, with a pointer to this MO discussion
created anti-reduced object, for completeness
Discussion with Mike reminded me that we were lacking an entry reflective subuniverse.
I started a template and cross-linked, but now I am out of battery and time before filling in any content. Will do that tomorrow.
some bare minimum at Chaitin’s incompleteness theorem
This is to flag up two entries that so-far just have titles. They are IulianUdrea and perfectly normal space. These may need watching. The second may be ok, and be a page somone has just started and intends to continue, but the name on the second also occurs as a name on a Mo page with no questions and no answers and may be someone seeing how many wikis etc they can put stuff on! Sorry for being a nasty suspicious b*****, but it looks a bit strange to me.
(N.B., the two entries do not seem to be related.)
at diffeomorphism I started listing theorems and references on statements about when the existence of a homeomorphism implies the existence of a diffeomorphism.
I dug out ancient references for the statement that in d≠4 everything homeomorphic to an open d-ball is also diffeomorphic to it. What would be a more modern, more canonical, more textbook-like reference for this?
And I’d also like to cite a reference for what is maybe obvious, that if that something in d=4 is an open subset of ℝ4 equipped with the induced smooth structure of the standard smooth structure, then the statement is also true in that dimension.
In fact, I am looking for nice/explicit/useful diffeomorphisms from the open n-ball onto the open n-simplex. I can of course fiddle around and cook up something, but I haven’t found anything that would count as nice. But probably some engineer out there working with finite elements or something does have a convenient choice.
there is this new master thesis:
which discusses aspects of weak Tarskian homotopy type universes following the indications that Mike Shulman has been making, for instance at universe (homotopytypetheory). I just got permission to share this and I have now included pointers to the thesis to that entry, to type of types, etc.
Created Vopenka’s principle.
At some point I had made up the extra axiom/terminology saying that an object 𝔸1 in a cohesive ∞-topos “exhibits the cohesion” if the shape modality is equivalent to 𝔸1-localization. Now I was talking about that assumption with Mike and noticed that this didn’t have a reflection on the nLab yet.
So now I have added, for the record, the definition here at “cohesive oo-topos” and cross-linked with the existing discussion at “continuum”.
added recent AlgTop mailing list contribution on fibrant replacement of cubical sets to cubical set