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am splitting off from Adams spectral sequence
for section 1.2.2 in Higher Algebra.
Is a generalized Eilenberg-MacLane space really, as the page says, a smash product of Eilenberg-MacLane spectra and not a wedge/cartesian product? In particular, in their appearance in the motivation of the Adams spectral sequence, it seems that from a finite number of cohomology classes $X\to \Sigma^{n_i}H \mathbb{F}_2$ one naturally obtains a map $X\to \prod_i \Sigma^{n_i}H \mathbb{F}_2$, not $X\to \bigwedge_i \Sigma^{n_i}H \mathbb{F}_2$.
I have created a list discrete and concrete objects - contents to be used as a “floating TOC” and have included it into relevant entries.
In this context I have also created concrete object, co-concrete object and discrete object and codiscrete object redirect to the existing discrete space and codiscrete space.
Added Qing Liu’s Algebraic Geometry and Arithmetic Curves - which is almost entirely self contained and which Werner Kleinert considers “the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form [after Grothendieck’s EGA]” - to the references of the arithmetic geometry page. On a trivial note: I didn’t forget the apostrophe in “Grothendieck’s”… it for some reason appears on the edit page but not on the actual nlab page.
have added to regular epimorphism the statement (here) that a pullback square of regular epis is also a pushout.
This must be true for effective epimorphisms in an $\infty$-topos, too. What’s the proof?
It may be silly, I know, but I wound up writing Pythagorean theorem. It’s not exactly “nPOV”. I did it to un-gray a link.
Created the page semi-topological K-theory.
created some bare minimum at symplectic spinors and metaplectic quantization
I have cross-linked the quadruple of entries
Euclidean geometry, Klein geometry
Riemannian geometry, Cartan geometry
and briefly edited the entries otherwise. For instance added an Idea-sentence to Euclidean geometry, and expanded Klein geometry (for instance the Examples).
Prompted by discussion with Thomas Holder and David Corfield behind the scenes, I have added some references:
Martin-Löf, A path from logic to metaphysics
Dieter Wandschneider, Dialektik als Letztbegründung der Logik
I’ve now created conic section.
I’ve added material to each of the entries polynomial ring, rational function, projective line, focusing mainly on the substitution operation and induced operations on the projective line.
stub for metaplectic group
I have started a page on identity among the relations to get rid of a grey link and to lead (eventually) to homotopy syzygy.
created a brief entry on quadratic Hamiltonians, just for completeness
am starting an entry on extended affine symplectic group, the restriction of the quantomorphism group of a symplectic vector space to elements that cover elements of the affine symplectic group.
(The name is not great, as the “extended” is too unspecific, but this seems to be close to standard in the literature – or else “extended inhomogeneous symplectic group”, which is not any better.)
stub for maximal torus
Somebody kindly points out the new preprint Morales-Zilber 14 to me, on Zilber’s program of combining model theory and physics. I have added the pointer there, but the entry remains a stub.
added to Haefliger groupoid some of the pertinent facts proven in Carchedi 12.
I have merged Mark Lawson with Mark V. Lawson as they are on the same person.
Why have all the orgins of changes suddenly become 127.0.0.1 i.e. local?
An extensive person entry Victor Snaith having some description of his 1979 Memoirs article (with a link to extended Russian translation). One of the aspects is an early version of an algebraic cobordism theory there, quite different both from Levine-Morel and from Voevodsky approach to alegbraic cobordism, and in a way to large. But the same construction is a special case of a more general metod in stable homotopy theory which may be of interest. Few words at algebraic cobordism as well. It would be nice to have more on Conner-Floyd isomorphism.
The reflection on the $n$Lab of the the topic cluster of tame geometric stacks – such as effective étale stacks, orbifolds/Deligne-Mumford stacks, their relation to pseudogroups, etc. – is still somewhat subotpimal. I’ll try to be editing a little in this direction.
For instance at proper geometric morphism there was no pointer to what is probably the main and original example, namely properness of diagonals for orbifold/DM-stacks. I have added at least one paragraph with pointers here. But eventually there should be some actual discussion.
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 $act\colon X \times Y \to Y$. This can be curried as $\hat{act}\colon Y \to Y ^ X$ where $Y ^ X$ 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 $Y ^ X$.
In the category Set there is no difference between the above left action and the right action $actR\colon Y \times X \to Y$ 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 $arrows\colon vertices \times labels \to vertices$ 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 $transition\colon inputs \times states \to states$. ↩
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 $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a compact Lie group, $T$ its maximal torus, $G^{\mathbb{C}}$ the complexification of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ 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?)
gave torsion of a Cartan connection its own entry, and cross-linked a bit.
Someone (anonymous) has raised a query at quality type. They say:
(Is “consists entirely of idempotents” correct?)
added to Cartan connection
definition
a standard reference
a standard example
Added pointers to Selberg zeta function for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function $\exp(i \pi \, \eta_D(0))$ 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 $n$Lab entries, so I ended up typing it right into the $n$Lab.
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 $\Gamma$ 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 $n \leq 6$ (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 $\pi_5(String(6)) = \mathbb{Z}$). 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 $H^6$ won’t be vanishing for oriented manifolds, one has some checking to do. For instance, the frame bundle of $S^6$ 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 $\widetilde{String(6)}$-bundle (i.e. the 6-connected cover of $String(6)$), since the transition function $S^5 \to BString(6)$ 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 $\mathbb{Z}$ to kill off in $String(8)$ to get $Fivebrane(8)$, so one gets a $U(1)\times U(1)$ higher gerbe. I suspect this larger $\pi_7$ 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 $\pi_3(SO(4)) = \mathbb{Z}\times \mathbb{Z}$ 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
I am giving fiber integration in K-theory a dedicated entry.
One section In operator K-theory used to be a subsection of fiber integration in generalized cohomology, and I copied it over.
Another section In terms of bundles of Fredholm operators I have now started to write.
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.