Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added doi-link to
created topological localization
brief category:people-entry for satisfying links now requested at p-adic Teichmüller theory
At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group π2(X,A) together with the bundary map δ:π2(X,A)→π1(A) and the π1(A)-action on π2(X,A). As someone confirms this example is correct I’ll add it to crossed module.
I am starting higher Segal space (while sitting in a talk by Mikhail Kapranov about them…)
expanded copower:
added an Idea-section, an Example-section, and a paragraph on copowers in higher category theory.
added the statement of the Fubini theorem for ends to a new section Properties.
(I wish this page would eventually give a good introduction to ends. I remember the long time when I banged my head against Kelly’s book and just didn’t get it. Then suddenly it all became obvious. It’s some weird effect with this enriched category theory that some of it is obvious once you understand it, but looks deeply mystifying to the newcomer. Kelly’s book for instance is a magnificently elegant resource for everyone who already understands the material, but hardly serves as an exposition of the ideas involved. I am hoping that eventually the nLab entries on enriched category theory can fill this gap. Currently they do not really. But I don’t have time for it either.)
starting an entry on the integer Heisenberg group.
For the moment it remains telegraphic as far as the text is concerned (no Idea-section)
but it contains a slick (I find) computation of the modular transformation of Chern-Simons/WZW states from the manifest modular automorphy of certain integer Heisenberg groups.
Hope to beautify this entry a little more tomorrow (but won’t have much time, being on an intercontinental flight) or else the days after (where I am however at a conference, but we’ll see).
a bare list of references, to be !include-ed into relevant entries (such as mapping class group and integer Heisenberg group) for ease of synchronization
created a currently fairly empty entry quantum measurement, just so as to have a place where to give a commented pointer to the article
[Reason for new thread: to all appearances, tricategory did not have one of its own, despite tetracategory having one]
(Updated reference to a representability theorem in arXiv:0711.1761v2 on tricategory; what was Theorem 21 in arXiv:0711.1761v1 has become Theorem 24 in arXiv:0711.1761v2 and its journal version)
a stub, right now just to satisfy links at classical double copy
More than half of this list is devoted to listing various proof assistants and formalization projects. Does this topic really warrant such an oversized representation in an article with a generic title “mathematics”?
Also, Categories and Sheaves, Sheaves in Geometry and Logic, Higher Topos Theory are good books, but do they really deserve such a prominent placement on top of the article? I suggest removing them.
Deleted broken links:
Theoretical Physics.Stack Exchange
research-level theoretical physics
basic and research-level Physics, and other STEM subjects
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.
Have added to HowTo a description for how to label equations
In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.
<div>
<p>created <a href="https://ncatlab.org/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a></p>
<p>the secret title of this entry is "Schreier theory done right". (where "right" is right from the <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a>)</p>
<p>this is the first part of the answer to</p>
<blockquote>
What is going on at <a href="https://ncatlab.org/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>?
</blockquote>
<p>The second part of the answer is the statement:</p>
<blockquote>
The same.
</blockquote>
<p>;-)</p>
<p>I'll expand on that eventually.</p>
</div>
a bare list of references on arguments
(by Connes) that Heisenberg’s original derivation of “matrix mechanics” and
more generally (by Ibort et al.) that Schwinger’s less known “algebra of selective measurements”
are both best understood, in modern language, as groupoid convolution algebras,
to be !include-ed into relevant entries (such as quantum observables and groupoid algebra), for ease of synchronizing
i polished the definition in bundle gerbe and then reorganized the former material on “Interpretations” in a new section
that first shows how to get a shifted central extension of groupoids form the bundle gerbe, and then demonstrates that this is the total space of a principal 2-bundle
I have expanded vertex operator algebra (more references, more items in the Properties-section) in partial support to a TP.SE answer that I posted here
a bare sub-section with a list of references, to be !include-ed into the References sections of relevant entries (such as at AQFT, FQFT and picture of mechanics)
I have expanded and edited moment map.
The induced map most likely isn’t a homeomorphism when X,Y are locally compact Hausdorff.
The original statement was in monograph by Postnikov without proof.
Not only that, in the current form it couldn’t possibly be true, since the map could lack to be bijective.
For more details see here: https://math.stackexchange.com/questions/3934265/adjunction-of-pointed-maps-is-a-homeomorphism .
I’ve added a reference in the case when X,Y are compact Hausdorff though.
Adam
Added:
Specifically, a continuous functor C→Set is a right adjoint functor if and only if it is representable, in which case the left adjoint functor Set→C sends the singleton set to the representing object
Created SVG Editor HowTo whilst trying to convert the codecogs monstrosity at exercise in groupoidification - the path integral to SVG (conversion happening in the Sandbox for the moment, and definitely not finished yet).
I tried to start an entry theta function, but it’s hard to tell for me if anything of it has been saved. The nLab is too busy doing something else than serving pages.
Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.
Added a reference.
Can we say exactly what kind of pretopos the category of small presheaves on a category C is?
Is it a ΠW-pretopos, provided that PC is complete?
As I’ve already said elsewhere, I’ve been working on this entry and trying to give a precise definition based on my hunches of what guys like Steenrod really meant by “a convenient category of topological spaces”. (I must immediately admit that I’ve never read his paper with that title. Of course, he meant specifically compactly generated Hausdorff spaces, but nowadays I think we can argue more generally.)
I also said elsewhere that my proposed axiom on closed and open subspaces might be up for discussion. The other axioms maybe not so much: dropping any of them would seem to be a deal-breaker for what an algebraic topologist might consider “convenient”. Or so I think.
Created categorical model of dependent types, describing the various different ways to strictify category theory to match type theory and their interrelatedness. I wasn’t sure what to name this page — or even whether it should be part of some other page — but I like having all these closely related structures described in the same place.
a stub entry, to give a home to today’s
stub for Poisson sigma-model. Needs references.
I have created an entry on the quaternionic Hopf fibration and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in G-equivariant stable homotopy theory it represents a non-torsion element in
[Σ∞GS7,Σ∞GS4]G≃ℤ⊕⋯for G a finite and non-cyclic subgroup of SO(3), and SO(3) acting on the quaternionic Hopf fibration via automorphisms of the quaternions.
I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.
Added the reference:
Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).
Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.