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tried to bring the entry Lie group a bit into shape: added plenty of sections and cross links to other nLab material. But there is still much that deserves to be done.
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.)
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
stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.
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 and of a vector space , we write if is finite-dimensional. We write and say and are commensurable if and .
A Tate vector space is a complete Hausdorff topological vector space that admits a basis of neighborhoods of 0 whose elements are mutually commensurable vector subspaces of .
A vector subspace of a Tate vector space is bounded if for every open vector subspace we have .
The dual of a Tate vector space is equipped with a topology generated by the basis of neighborhoods of 0 whose elements are orthogonal complements to bounded subspaces of .
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
Created an entry for this.
I’ve adopted the existing convention at nLab in the definition of (which is also the definition I prefer).
Since the opposite convention is used a lot (e.g. by Lurie), I’ve decided it was worth giving it notation, the relation between the versions, and citing results in both forms. Since I didn’t have any better ideas, I’ve settled on .
category: people page for the reference
Anonymouse
At coverage, I just made the following change: Where the sheaf condition previously read
it now uses the variable names “” and “” instead of “” and “”:
I’m announcing this almost trivial change because I’d like to invite objections, in which case I’d rollback that change and also would not go on to copy this change to related entries such as sheaf. There are two tiny reasons why I prefer the new variable names:
The entry (infinity,1)-Kan extension is still a sad stub which you shouldn’t look at if you have better things to do. But I have now briefly added at least a few more specific pointers to HTT, in particular to the pointwise-ness issue. But just pointers, essentially no text for the moment. (If you feel energetic, be invited to turn the entry into something prettier!)
For completeness I have added pointer to
though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves.
Adding reference
Anonymouse
added pointer to:
Zhiyuan Wang, Kaden R. A. Hazzard: Particle exchange statistics beyond fermions and bosons, Nature 637 (2025) 314-318 [arXiv:2308.05203, doi:10.1038/s41586-024-08262-7]
Zhiyuan Wang: Parastatistics and a secret communication challenge [arXov:2412.13360]
making this a stand-alone entry (“2-sphere” used to redirect to sphere, which however ended up being about -spheres in generality)
but it is just a stub for the time being. Mainly I was looking to make a home for these references on :
in relation to braid groups:
and regarded as a classifying space, (for “ine” bundles):
Jack Morava: A homotopy-theoretic context for CKM/Birkhoff renormalization [arXiv:2307.10148, spire:2678618]
Jack Morava: Some very low-dimensional algebraic topology [arXiv:2411.15885]