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added this pointer on the homotopy groups of the embedded cobordism category:
Marcel Bökstedt, Anne Marie Svane, A geometric interpretation of the homotopy groups of the cobordism category, Algebr. Geom. Topol. 14 (2014) 1649-1676 (arXiv:1208.3370)
Marcel Bökstedt, Johan Dupont, Anne Marie Svane, Cobordism obstructions to independent vector fields, Q. J. Math. 66 (2015), no. 1, 13-61 (arXiv:1208.3542)
Anel and Catren in their introduction to New Spaces in Physics claim that Lagrangian submanifolds are category-theoretic “points” of a symplectic manifold, morphisms from the trivial symplectic manifold in Weinstein’s symplectic category.
Is this accurate?
Anonymous
The entry on topological group could stand more work, but I added some stuff on the uniform structure, in particular the proposition that for group homomorphisms , continuity at a single point guarantees uniform continuity over all of . The proof is follow-your-nose, of course.
What we really need is an entry Haar measure. I’ll get started on that soon.
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:
edited retract a little
In discrete fibration I added a new section on the Street’s definition of a discrete fibration from to , that is the version for spans of internal categories. I do not really understand this added definition, so if somebody has comments or further clarifications…
created (finally) lax monoidal functor (redirecting monoidal functor to that) and strong monoidal functor.
Hope I got the relation to 2-functors right. I remember there was some subtlety to be aware of, but I forget which one. I could look it up, but I guess you can easily tell me.
Todd,
when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?
Thanks!
starting a category:reference
-entry.
Just a single item so far, but this entry should incrementally grow as more preprints appear (similar to what we have been doing at Handbook of Quantum Gravity and similar entries).
I know that a soft deadline for submissions of at least one of the sections is this December, so I am guessing this is planned to appear in 2024.
stub for von Neumann algebra factor
I’ve made a few changes at flat module since I wanted to know what one was and the nLab page simply confused me further. It seemed to be saying that a module over a ring/algebra is flat if tensoring with is a flat functor. That seemed absurd so I changed it. The observation “everything happens for a reason” was a little curt so once I’d worked out what it meant I expanded it a bit and put in the analogy to bases. It wasn’t clear from the way that it was phrased whether this condition was due to Wraith and Blass or the fact that it can be put in a more general context. Lastly, the last sentence was originally in the same paragraph as the penultimate sentence where it didn’t seem to fit (or was at best ambiguous) and it also claimed that the module could be non-unital which seemed a little odd.
If an expert could kindly check that I’ve done no lasting damage to the page, I’d be grateful.
added pojnter to Maldacena-Nunez 01
I have touched the Idea-section at first-order formulation of gravity, trying to improve a little.
considerably expanded the entry strict 2-group.
Apart from adding an introductory discussion, and expanding the list of examples, in particular by adding that of automorphism 2-groups ...
... I in particular give the detailed translation prescription for how to encode a 2-group by a crossed module at In terms of crossed modules
This is to eventually serve as a supplement to the discussion at nonabelian group cohomology. So I spent some energy on disentangling the four different (though isomorphic) ways a crossed module gives rise to a 2-group (following my article with David Roberts).
starting a stand-alone Section-entry (to be !include
ed as a section into D=11 supergravity and into D’Auria-Fré formulation of supergravity)
So far it contains lead-in and statement of the result, in mild but suggestive paraphrase of CDF91, §III.8.5.
I am going to spell out at least parts of the proof, with some attention to the prefactors.
I edited
to make it clear that this tensor product only works for finite abelian categories, which are what we get from looking at finite-dimensional representations of finite-dimensional associative algebras. It’s all very finite… and over fields, too, at least in the treatment by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, which is all I have access to now. It would be nice to see this kind of thing done more generally. The slides by Ignacio López Franco, linked on this page, are a hint of how to do it.
Created bare minimum for orthocompact, which will be expanded later.
Edit: I’m not sure if there is a convention for the title? With or without “topological”? One one hand there are compact space, metacompact space or hemicompact space. On the other hand, there are locally compact topological space or paracompact topological space.
Edit: I’ve now linked orthocompace space on hemicompact space and paracompact topological space as there are lemmata connecting them with each other.
started Weinstein symplectic category
Created bare minimum for rational homology sphere, which will be expanded later.
added pointer to today’s:
Added Mazurkiewicz theorem and Hausdorff Gδ theorem with original reference by Stefan Mazurkiewicz and Felix Hausdorff. Linked english Wikipedia page. (The german Wikipedia page is now also available.)
added the pointers to the combinatorial proofs of the fiberwise detection of acyclicity of Kan fibrations, currently discussed on the AlgTop list, to the nLab here.
at vector bundle I have spelled out the proof that for paracompact Hausdorff then the restrictions of vector bundles over to and are isomorphic.
It’s just following Hatcher, but I wanted to give full detail to the argument of what is now this lemma.
Created page for P-space. (I intend to create the page for orthocompact spaces in the future. The german Wikipedia page is now also available.)
Created page for Stefan Mazurkiewicz with reference of the Mazurkiewicz theorem both featured on G-delta subspace.
crated D'Auria-Fre formulation of supergravity
there is a blog entry to go with this here
added this quote to before the Idea-section:
In the wake of the movement of ideas which followed the general theory of relativity, I was led to introduce the notion of new geometries, more general than Riemannian geometry, and playing with respect to the different Klein geometries the same role as the Riemannian geometries play with respect to Euclidean space. The vast synthesis that I realized in this way depends of course on the ideas of Klein formulated in his celebrated Erlangen programme while at the same time going far beyond it since it includes Riemannian geometry, which had formed a completely isolated branch of geometry, within the compass of a very general scheme in which the notion of group still plays a fundamental role.
[Élie Cartan 1939, as quoted in Sharpe 1997, p. 171]
added pointer to:
here and elsewhere
A generalization of Waldhausen K-theory to dualizable dg-categories and dualizable stable ∞-categories.
For compactly generated inputs, recovers the Waldhausen K-theory of the full subcategory of compact objects.
The formalism is applicable to -presentable stable ∞-categories, where can be uncountable (for example, various categories of sheaves, or categories occurring in functional analysis).
Alexander Efimov, On the K-theory of large triangulated categories, ICM 2022, https://www.youtube.com/watch?v=RUDeLo9JTro
Marc Hoyois, K-theory of dualizable categories (after A. Efimov), https://hoyois.app.uni-regensburg.de/papers/efimov.pdf.
Li He, Efimov K-theory and universal localizing invariant, arXiv:2302.13052.
added to string theory FAQ two new paragraphs:
Prompted by the MO discussion
I reorganized linearly distributive category by moving the long block of history down to the bottom, adding an “Idea” section and a description of how -autonomous categories give rise to linearly distributive ones and linearly distributive ones give rise to polycategories. I also cross-linked the page better with polycategory and star-autonomous category.
started topologically twisted D=4 super Yang-Mills theory, in order to finally write a reply to that MO question we were talking about. But am being interrupted now…
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.
added some minimum of content to this stub entry, including pointer to today’s