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Created an entry for this.
I’ve adopted the existing convention at nLab in the definition of Tw(C) (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 ¯Tw(C).
category: people page for the reference
Anonymouse
At coverage, I just made the following change: Where the sheaf condition previously read
X(U)→∏i∈IX(Ui)⇉∏i,j∈IX(Ui×UUj),it now uses the variable names “j” and “k” instead of “i” and “j”:
X(U)→∏i∈IX(Ui)⇉∏j,k∈IX(Uj×UUk).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 n-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 ΩS2:
in relation to braid groups:
and regarded as a classifying space, ΩS2≃BΩ2S2 (for “line” 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]
created a bare minimum at light-cone gauge quantization, just so as to be able to sensibly link to it from elsewhere
started Thom space
added pointer to these two recent references, identifying further L∞-algebra structure in Feynman amplitudes/S-matrices of perturbative quantum field theory:
Markus B. Fröb, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories (arXiv:1803.10235)
Alex Arvanitakis, The L∞-algebra of the S-matrix (arXiv:1903.05643)
I am starting higher spin gauge theory
Added link to finite étale morphism of anabelioids.
Started the stub for semilinear map. More to come.
added these two quotes:
Yang wrote in C. N. Yang, Selected papers, 1945-1980, with commentary, W. H. Freeman and Company, San Francisco, 1983, on p. 567:
In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of S. S. Chern in El Cerrito, near Berkeley… I said I found it amazing that gauge theory are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added: “this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.” He immediately protested: “No, no. These concepts were not dreamed up. They were natural and real.
Yang expanded on this passage in an interview recorded as: C. N. Yang and contemporary mathematics, chapter in: Robin Wilson, Jeremy Gray (eds.), Mathematical Conversations: Selections from The Mathematical Intelligencer, Springer 2001, on p. 72 (GoogleBooks):
But it was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the structure of the physical world is intimately tied to the deep mathematical concepts, concepts which were developed out of considerations rooted only in logic and the beauty of form?
Started an entry in “category:motivation” on fiber bundles in physics.
(prompted by this Physics.SE question)
I removed some spam on category theory.
recording the 1-categorical equivalence Ho(CombModCat)≃Ho(PresentableDerivators) obtained from Renaudin06
Created stub. This used to redirect to codomain fibration, but I think that’s wrong.
I have expanded various sections at disjoint coproduct. In particular towards the end is now a mentioning of the fact that in a positive category morphisms into a disjoint coproduct are given by factoring disjoint summands of the domain through the canonical inclusions.
Also,I made positive category and variants redirect to extensive category.
a bare list of references, to be !include-ed into lists of references of relevant entries (such as 2d CFT, 2d SCFT, conformal cobordism category, modular functor and maybe elsewhere)
at DHR superselection theory I have added the argument (here) for why every DHR representation indeed comes from a net-endomorphism, assuming Haag duality and that the net takes values in vN algebras.