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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added at Grothendieck universe at References a pointer to the proof that these are sets of κ-small sets for inaccessible κ. (also at inaccessible cardinal)
Together with my PhD students, I have been thinking a lot recently about the appropriate notion of a module over a C^∞-ring, i.e., something with better properties than Beck modules, which boil down to modules over the underlying real algebra in this case.
We stumbled upon the article C-infinity module (schreiber).
It says: “a C-infty algebra A is a copresheaf A∈Quantities=CoPrSh(CartesianSpaces) which becomes a copresheaf with values in algebras when restricted along FinSet↪CartesianSpaces,”
Why are we restricting to FinSet here? The underlying commutative real algebra is extracted by restricting to the Lawvere theory of commutative real algebras, i.e., CartesianSpaces_Poly, the subcategory of cartesian spaces and polynomial maps. Restricting to FinSet^op (as opposed to FinSet) extracts the underlying set only. It is unclear what is being meant by restricting along FinSet→CartesianSpaces, since the latter functor does not preserve finite products, so restricting along it does not produce a functor between categories of algebras over Lawvere theories.
In this entry, generating functional redirects to generating function. This use does not seem to match the context.
some minimum, for the moment mostly to record this item:
added some formatting and some cross-links (nilpotent groups!) and added pointer to:
stub at locally compact locale
Added:
Introducing hyperstonean spaces:
Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.
splitting this off from su(2)-anyons: Copied much of the material over, but also added a few more sentences.
For the moment this entry is a cautionary tale about confirmation bias more than an entry about physics.
am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.
started a stubby double dimensional reduction in this context and added some first further pointers and references to M5-brane, to D=5 super Yang-Mills theory and maybe elsewhere.
But this still needs more details to be satisfactory, clearly.
Just noticed that we have a duplicate page Jon Sterling.
I have now moved the (little but relevant) content (including redirects) from there to here.
Unfortunately, the page rename mechanism seems to be broken until further notice, therefore I am hesitant to clear the page Jon Sterling completely, for the time being.
I looked at real number and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:
A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, denoted ℝ. The underlying set is the completion of the ordered field ℚ of rational numbers: the result of adjoining to ℚ suprema for every bounded subset with respect to the natural ordering of rational numbers.
The set of real numbers also carries naturally the structure of a topological space and as such ℝ is called the real line also known as the continuum. Equipped with both the topology and the field structure, ℝ is a topological field and as such is the uniform completion of ℚ equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces ℝn for natural numbers n∈ℕ – the real line ℝ is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.
category: people page for the reference
Anonymouse
added hyperlinks to the text at induced representation. Made sure that it is cross-linked with Frobenius reciprocity.
Stub Frobenius reciprocity.
stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.
I added the reference
a stub entry, for the time being just to satisfy a link that has long been requested at Quillen Q-construction
added pointer to:
created black holes in string theory, since somebody asked me: a brief paragraph explaining how the entropy-counting works and some references.
have created a stub for supersymmetric quantum mechanics
Zoran, I see that you once dropped a big query box at quantum mechanics with a complaint. I disagree with the point you make there: we have fundamental definitions of quantum field theory and restricting them to 1 dimension gives quantum mechanics. If you want to turn this around and understand all QFTs as infinite-dimensional quantum mechanics (which, yes, one can do) you are discarding the nice conceptual models and kill the concept of extended QFT.
In any case, I think remarks like this (in the style of “we can also regard this the other way round like this”) are better added into an entry as what they are – remarks – than as query boxes that give the impression that there is something fishy about the rest of the entry.
I came across this tiny page which is called by ’extension’ from the page central extension.
But what’s this page trying to be? Merely about a certain kind of field extension?
Created:
[…]
Introduced by
Survey:
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.
stub for jet bundle
was initially just looking for a page to host this reference:
but now I wrote a little bit of an Idea-section, too. Leaving much room to be further expanded, of course.
made some cosmetic adjustments to the entry,
and slightly expanded the last remark (which I made a Remark) relating to order theory (prodded by this comment)
Added a budget of links from the FAQ to the Eunuch-Code Data-Bank.
For some odd reason, the TOC generator is not picking up the first heading — ???
I added a synthetic definition of open subspace due to Penon.
Created:
An intrinsic notion of an open subobject in an elementary topos.
A monomorphism U→X in an elementary topos E is a Penon open if the following statement holds in the internal logic of E:
∀x∈X∀y∈X(x∈U)→(¬(y=x)∨y∈U).If U→X is a Penon open, then
∀x∈U({y∈X∣¬¬(y=x)}⊂U).Jacques Penon, De l’infinitésimal au local (Thèse de Doctorat d’État), Diagrammes S13 (1985), 1-191. numdam.
Eduardo J. Dubuc, Jacques Penon, Objets compacts dans les topos, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40:2 (1986), 203-217. doi.
Jacques Penon, Infinitésimaux et intuitionnisme, Cahiers de topologie et géométrie différentielle 22:1 (1981), 67-72. numdam.
Oscar P. Bruno, Logical opens of exponential objects, Cahiers de Topologie et Géométrie Différentielle Catégoriques 26:3 (1985), 311-323.
Marta C. Bunge, Felipe Gago, Ana María San Luis, Synthetic Differential Topology, Cambridge University Press, 2018. ISBN: 9781108553490, DOI.
Numerous bibliographic additions to linguistics and new stub mathematical linguistics.
added pointer to: