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Created.
Removed a query:
+– {: .query} Bruce: I’m shooting in the dark here with this ω-groupoid sentence above. Am I right? What does that boil down to concretely? =–
+– {: .query}
Bruce: What’s a geometric stack?
Chris: A stack is geometric if it is quasi-compact (any open cover has a finite sub-cover) and the diagonal morphism is representable and affine, though that probably doesn’t help much. I don’t know much about stacks yet, but maybe someone else can explain this. I think the point is that one needs some hypotheses to actually prove stuff for stacks.
=–
Created homotopy equivalence of toposes.
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.
Is this page a duplicate of Jiří Vanžura?
brief category:people-entry for hyperlinking references at quaternion-Kähler manifold, quaternion-Kähler manifold and elsewhere
There seem to be some misleading remarks at Čech model structure on simplicial presheaves.
Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.
Marc Hoyois seems to says the opposite: there is no deep relation between “Čech” in “Čech cohomology” and in “Čech model structure”.
[…] the corresponding Čech cover morphism .
Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.
The Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don’t know how to prove it otherwise. Of course, injective-cofibrancy is trivial.
this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be
Based on the discussion here, it seems that the Čech model structure is not site-independent, even though it can be defined on the category of simplicial sheaves. A very strange state of affairs…
Added these pointers to today’s replacements:
Changha Choi, Leon A. Takhtajan: Supersymmetry and trace formulas I. Compact Lie groups [arXiv:2112.07942]
Changha Choi, Leon A. Takhtajan: Supersymmetry and trace formulas II. Selberg trace formula [arXiv:2306.13636]
I added some more to Lebesgue space about the cases where 1<p<∞ fails.
Added a bunch of material to inverse semigroup under subsections of “Properties”.
I have added some discussion to variable.
Not sure if I ever announced this here, the original version is a few months old already. But right now I have added also some sentences on bound variables.
Some more professional logician might please look over this. There will be lots of room to improve on those few sentences that I jotted down, mainly such as to have something there at all.
a stub, to make links work
(This used to be a stub “quantum circuit” which I just quasi-duplicated at a more extensive entry quantum circuit diagram. But since quantum gate was already redirecting here – which is how I discovered/remembered that this entry exists – no harm is done by making that it’s new title.)
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)
The link for ’equivalent’ at the top redirected to natural isomorphism which (as I understand it) is the correct 1-categorical version of an equivalence of functors, but this initially lead me to believe that a functor was monadic iff it was naturally isomorphic to a forgetful functor from the Eilenberg-Moore category of a monad on its codomain, which would mean that the domain of the functor was literally the Eilenberg-Moore category of some adjunction since natural isomorphism is only defined for parallel functors.
gave the statement that derivations of smooth functions are vector fields a dedicated entry of its own, in order to be able to convieniently point to it
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.)
created stub entry for double nerve in reply to this MO question.
Several recent updates to literature at philosophy, the latest being
which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…
Created:
Given vector subspaces V0 and V1 of a vector space V, we write V0≺V1 if V0/(V0∩V1) is finite-dimensional. We write V0∼V1 and say V0 and V1 are commensurable if V0≺V1 and V1≺V0.
A Tate vector space is a complete Hausdorff topological vector space V that admits a basis of neighborhoods of 0 whose elements are mutually commensurable vector subspaces of V.
A vector subspace W of a Tate vector space V is bounded if for every open vector subspace U⊂V we have W≺U.
The dual of a Tate vector space V is Hom(V,C) equipped with a topology generated by the basis of neighborhoods of 0 whose elements are orthogonal complements to bounded subspaces of V.
Tate vector spaces form an pre-abelian category.
John Tate, Residues of differentials on curves, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 1 (1968) no. 1, pp. 149-159. DOI.
Alexander Beilinson, Boris Feigin, Barry Mazur, Notes on conformal field theory. PDF
commented in the discussion at point of a topos and have a question there.
have expanded the Idea-section at quantum logic to now read as follows:
Broadly speaking, quantum logic is meant to be a kind of formal logic that is to traditional formal logic as quantum mechanics is to classical mechanics.
The first proposal for such a formalization was (Birkhoff-vonNeumann 1936), which suggested that given a quantum mechanical system with a Hilbert space of states, the logical propositions about the system are to correspond to (the projections to) closed subspaces, with implication given by inclusion of such subspaces, hence that quantum logic is given by the lattice of closed linear subspaces of Hilbert spaces.
This formalization is often understood as being the default meaning of “quantum logic”. But the proposal has later been much criticized, for its lack of key properties that one would demand of a formal logic. For instance in (Girard 11, page xii) it says:
Among the magisterial mistakes of logic, one will first mention quantum logic, whose ridiculousness can only be ascribed to a feeling of superiority of the language – and ideas, even bad, as soon as they take a written form – over the physical world. Quantum logic is indeed a sort of punishment inflicted on nature, guilty of not yielding to the prejudices of logicians… just like Xerxes had the Hellespont – which had destroyed a boat bridge – whipped.
and for more criticisms see (Girard 11, section 17).
Therefore later other proposals as to what quantum logic should be have been made, and possibly by “quantum logic” in the general sense one should understand any formal framework which is supposed to be able to express the statements whose semantics is the totality of all what is verifiable by measurement in a quantum system.
In particular it can be argued that flavors of linear logic and more generally linear type theory faithfully capture the essence of quantum mechanics (Abramsky-Duncan 05, Duncan 06, see (Baez-Stay 09) for an introductory exposition) due to its categorical semantics in symmetric monoidal categories such as those used in the desctiption of finite quantum mechanics in terms of dagger-compact categories. In particular the category of (finite dimensional) Hilbert spaces that essentially underlies the Birkhoff-vonNeumann quantum logic interprets linear logic.
Another candidate for quantum logic has been argued to be the internal logic of Bohr toposes .
In quantum computing the quantum analog of classical logic gates are called quantum logic gates.
I added some material to Peano arithmetic and Robinson arithmetic. At the latter, I replaced the word “fragment” (which sounds off to my ears – actually Wikipedia talks about thisterm a little) with “weakening”.
Still some links to be inserted.
Included the condition on sequential (co)limits that the indexing ordinal should be nonzero, which I presume to be the correct convention. (e.g. based on the description they are a special case of filtered colimits)
splitting article on sequential limits into two articles, on sequential limits and sequential colimits
Anonymouse