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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added hyperlinks to the text at induced representation. Made sure that it is cross-linked with Frobenius reciprocity.
Created a stub entry for norm map, for the moment just so as to make cross-links work.
I made some changes to bivector. While the idea section is correct (and should be strictly adhered to!) but the previous definition is wrong in general! The previous definition is consistent and used in wikipedia but it misses both the direct relation of bivectors, trivectors and general polyvectors to determinants as well as the standard nontrivial usage of bivectors in analytic geometry wher bivectors define equivalence classes of parallelograms and in particular with a point in space given define an affine plane. If we adhere to wikipedia and not to standard treatments in geometry (e.g. M M Postnikov, Analytic geometry) then we miss the nontriviality of the notion of bivector and its meaning which is more precise than that of a general element in the second exterior power.
Bivector in a vector space V is not any element in the second exterior power, but a DECOMPOSABLE vector in the second tensor power – in general dimension just such elements in Λ2V have the intended geometric meaning and define vector 2-subspaces and of course affine 2-subspaces if a point in the 2-subspace is given. It is true that every bivector in 2-d or in 3-d space is decomposable, but in dimension 4 this is already not true. Thus the bivectors form a vector space just in the dimensions up to 3. Similarly, trivectors form a vector space just in the dimensions up to 4. In the context of differential graded algebras, polyvector fields are usually taken as arbitrary elements in the exterior powers of vector fields.
I am trying to collect citable/authorative references that amplify the analog of the mass gap problem in particle phenomenology, where it tramslates into the open problem of computing hadron masses and spins from first principles (due to the open problem of showing existence of hadrons in the first place!).
This is all well and widely known, but there is no culture as in mathematics of succinctly highlighting open problems such that one could refer to them easily.
I have now created a section References – Phenomenology to eventually collect references that come at least close to making this nicely explicit. (Also checked with the PF community here)
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.
Wrote an article Eudoxus real number, a concept due to Schanuel.
added to the Properties-section at Hopf algebra a brief remark on their interpretation as 3-vector spaces.
I came to think that the term geometric type theory for the type theory internal toi sheaf toposes should exists. Thanks to Bas Spitter for pointing out that Steve Vickers had already had the same idea (now linked to at the above entry).
Also created geometric homotopy type theory in this vein, with some evident comments.
I have expanded slightly at coalgebra – Properties – As filtered colimits of finite dimensional pieces.
And I have added and cross-linked with corresponding remarks at dg-coalgebra, at pro-object, at L-infinity algebra and at model structure for L-infinity algebras.
https://ncatlab.org/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves
In the article here there is a confusing error in remark 2.2: in the diagram, A is contravariant, so that either the number of arrows must change or the diagram must be reversed. For comparison, the article
https://ncatlab.org/nlab/show/(infinity,1)-sheaf
in proposition 2.1 gets essentially the same correct.
following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be !include-ed into the References sections of relevant entries (mainly at homotopy theory and at algebraic topology) for ease of updating and syncing these lists.
The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.
I’ll just check now that I have all items copied, and then I will !include this entry here into homotopy theory and algebraic topology. It may best be viewed withing these entries, because there – but not here – will there be a table of contents showing the subsections here.
This is a bare list of references, to be !include-ed into the References-lists of relevant entries (such as at anyon, topological order, fusion category, unitary fusion category, modular tensor category).
There is a question which I am after here:
This seems to be CMT folklore, as all authors state it without argument or reference.
Who is really the originator of the claim that anyonic topological order is characterized by certain unitary braided fusions categories/MTCs?
Is it Kitaev 06 (which argues via a concrete model, in Section 8 and appendix E)?
Created:
In algebraic geometry, the module of Kähler differentials of a commutative ring R corresponds under the Serre–Swan duality to the cotangent bundle of the Zariski spectrum of R.
In contrast, the module of Kähler differentials of the commutative real algebra of smooth functions on a smooth manifold M receives a canonical map from the module of smooth sections of the cotangent bundle of M that is quite far from being an isomorphism.
An example illustrating this point is M=R, since in the module of (traditionally defined) Kähler differentials of C∞(M) we have d(exp(x))≠expdx, where exp:R→R is the exponential function. That is to say, the traditional algebraic notion of a Kähler differential is unable to deduce that exp′=exp using the Leibniz rule.
However, this is not a defect in the conceptual idea itself, but merely a failure to use the correct formalism. The appropriate notion of a ring in the context of differential geometry is not merely a commutative real algebra, but a more refined structure, namely, a C^∞-ring.
This notion comes with its own variant of commutative algebra. Some of the resulting concepts turn out to be exactly the same as in the traditional case. For example, ideals of C^∞-rings and modules over C^∞-rings happen to coincide with ideals and modules in the traditional sense. Others, like derivations, must be defined carefully, and definitions that used to be equivalent in the traditional algebraic context need not remain so in the context of C^∞-rings.
Observe that a map of sets d:A→M (where M is an A-module) is a derivation if and only if for any real polynomial f(x1,…,xn) the chain rule holds:
d(f(a1,…,an))=∑i∂f∂xi(x1,…,xn)dxi.Indeed, taking f(x1,x2)=x1+x2 and f(x1,x2)=x1x2 recovers the additivity and Leibniz property of derivations, respectively.
Observe also that f is an element of the free commutative real algebra on n elements, i.e., R[x1,…,xn].
If we now substitute C^∞-rings for commutative real algebras, we arrive at the correct notion of a derivation for C^∞-rings:
A __C^∞-derivation__ of a [[C^∞-ring]] $A$ is a map of sets $A\to M$ (where $M$ is a [[module]] over $A$) such that the following chain rule holds for every smooth function $f\in\mathrm{C}^\infty(\mathbf{R}^n)$:
$$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i,$$
where both sides use the structure of a [[C^∞-ring]] to evaluate a smooth real function on a collection of elements in $A$.
The module of Kähler C^∞-differentials can now be defined in the same manner as ordinary Kähler differentials, using C^∞-derivations instead of ordinary derivations.
\begin{theorem} (Dubuc, Kock, 1984.) The module of Kähler C^∞-differentials of the C^∞-ring of smooth functions on a smooth manifold M is canonically isomorphic to the module of sections of the cotangent bundle of M. \end{theorem}
The book is no longer in progress, but published 8 years ago. I added the detail and a link to the AMS page.
I think it would be good to include a paragraph on the claims in the book about (∞,2)-categories the authors explicitly say they don’t prove and can’t find a proof in the literature. Just flagging this for now. Ultimately, when the papers finishing the proofs of these claims land on the arXiv, these can be cited.
Added:
\tableofcontents
A model for monoidal (∞,1)-categories.
A monoidal relative category is a monoidal category equipped with a relative category structure such that the monoidal product preserves weak equivalences.
The canonical functor from the quasicategorical localization of the relative category of monoidal relative categories, monoidal relative functors, and monoidal Dwyer–Kan equivalences to the quasicategory of monoidal quasicategories is a weak equivalence.
I gave André Joyal’s lectures in Paris last week their own category:reference page on the nLab, in order to be able to link to them conveniently (from entries such as topos theory and (infinity,1)-topos theory):
started a stub for ambidextrous adjunction, but not much there yet
Added
I started a minimal entry at categorical geometric Langlands conjecture, which is supposed to be about the Arinkin-Gaitsgory reformulation of the geometric Langlands conjecture as an equivalence between stable infinity-categories.
I also created pages for the references Notes on geometric Langlands and A study in derived algebraic geometry.
a beginning at geometric Langlands correspondence
Mike Stay kindly added the standard QM story to path integral.
I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.
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.