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    • For now creating page, more to be added soon.

      v1, current

    • while bringing some more structure into the section-outline at comma category I noticed the following old discussion there, which hereby I am moving from there to here:

      [begin forwarded discussion]

      +–{.query} It's a very natural notation, as it generalises the notation (x,y)(x,y) (or [x,y][x,y] as is now more common) for a hom-set. But personally, I like (f→g)(f \rightarrow g) (or (f↘g)(f \searrow g) if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from ff to gg. —Toby Bartels

      Mike: Perhaps. I never write (x,y)(x,y) for a hom-set, only A(x,y)A(x,y) or hom A(x,y)hom_A(x,y) where AA is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen [x,y][x,y] for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.

      I would be okay with calling the comma category (or more generally the comma object) E(f,g)E(f,g) or hom E(f,g)hom_E(f,g) if you are considering it as a discrete fibration from AA to BB. But if you are considering it as a category in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer (f/g)(f/g) as less visually distracting, and evidently a generalization of the common notation C/xC/x for a slice category.

      Toby: Well, I never stick ‘EE’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.

      Mike: The main reason I don’t like unadorned (f,g)(f,g) for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see (f,g)(f,g) in a category is that we have f:X→Af:X\to A and g:X→Bg:X\to B and we’re talking about the pair (f,g):X→A×B(f,g):X\to A\times B — surely also a natural generalization of the very well-established notation for ordered pairs.

      Toby: The notation (f/g/h)(f/g/h) for a double comma object makes me like (f→g→h)(f \to g \to h) even more!

      Mike: I’d rather avoid using →\to in the name of an object; talking about projections p:(f→g)→Ap:(f\to g)\to A looks a good deal more confusing to me than p:(f/g)→Ap:(f/g)\to A.

      Toby: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If f,g:A→Bf, g: A \to B, then f→gf \to g ought to be the set of transformations between them. (Or f⇒gf \Rightarrow g, but you can't keep that decoration up.)

      Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation (f,g)(f,g) is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation (f,g)(f,g) for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of ff and gg are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the fibers of the comma category, considered as a fibration from CC to DD, that are hom-sets. Finally, I don’t think the notation (f,g)(f,g) scales well to double comma objects; we could write (f,g,h)(f,g,h) but it is now even less like a hom-set.

      Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use M[C→fE←gD]M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]. Maybe comma[C→fE←gD]comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]? Lengthy, but at least unambiguous. Or maybe fE I g{}_f {E^I}_g?

      Zoran Skoda: (f/g)(f/g) or (f↓g)(f\downarrow g) are the only two standard notations nowdays, I think the original (f,g)(f,g) which was done for typographical reasons in archaic period is abandonded by the LaTeX era. (f/g)(f/g) is more popular among practical mathematicians, and special cases, like when g=id Dg = id_D) and (f↓g)(f\downarrow g) among category experts…other possibilities for notation should be avoided I think.

      Urs: sounds good. I’ll try to stick to (f/g)(f/g) then.

      Mike: There are many category theorists who write (f/g)(f/g), including (in my experience) most Australians. I prefer (f/g)(f/g) myself, although I occasionally write (f↓g)(f\downarrow g) if I’m talking to someone who I worry might be confused by (f/g)(f/g).

      Urs: recently in a talk when an over-category appeared as C/aC/a somebody in the audience asked: “What’s that quotient?”. But (C/a)(C/a) already looks different. And of course the proper (Id C/const a)(Id_C/const_a) even more so.

      Anyway, that just to say: i like (f/g)(f/g), find it less cumbersome than (f↓g)(f\downarrow g) and apologize for having written (f,g)(f,g) so often.

      Toby: I find (f↓g)(f \downarrow g) more self explanatory, but (f/g)(f/g) is cool. (f,g)(f,g) was reasonable, but we now have better options.

      =–

    • brief category:people-entry for hyperlinking references

      v1, current

    • The first paragraph had a link to “fiber functor”, which takes one to the very same page.

      diff, v8, current

    • How would people feel about renaming distributor to profunctor? I seem to recall that when this came up on the Cafe, I was the main proponent of the former over the latter, and I've since changed my mind.

    • added to supergeometry a link to the recent talk

      • Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres (video)

    • A stub, for the moment just to have a place for recording a couple of references (which were previously at fusion category.

      v1, current

    • At closed subspace, I added some material on the 14 operations derivable from closures and complements. For no particularly great reason except that it’s a curiosity I’d never bothered to work through until now.

    • starting page on σ\sigma-pretopological spaces

      Anonymouse

      v1, current

    • for ContinuousFunctionsDetectedByNets, the backwards direction had an indirect proof. I added a direct one.

      Anonymous

      diff, v37, current

    • starting page on unified topologies

      Anonymouse

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Copy-pasted the very explicit definition added to the page “monoidal category”.

      diff, v7, current

    • Added link to “multiplicatively cancellable semi-ring”.

      diff, v32, current

    • Added a simpler example of a nontopological pretopological space, modified from a remark by Mike Shulman.

      diff, v11, current

    • I tried to prettify the entry topological space a bit more:

      • made an attempt at adding an Idea-section (feel free to work on that, it’s just a quick idea motivated more from the desire to have such a section at all than from an attempt to do it any justice).

      • collected the three Definition-sections to subsections of a single Definition-section

      • polished and expanded the Standard definition section.

    • Wikipedia has a nice article on quantum operations.

      The nLab also had a page quantum operations and channels (cache bug?), but I’ve renamed this to simply quantum operation since a quantum channel seems to be nothing but a quantum operation when viewed from the perspective of quantum information theory. Eventually, this page might need some disambiguation since there may be several uses of the term, but for now I think it is “ok”.

      I think this page can be cleaned up. I started, but don’t think I will be able to finish.

      In particular, there is some background material that might be better on separate pages. I’ll continue trying to clean things up, but family might be calling soon and I’ll need to run quickly whatever state it is in.

      I also made the simple statement

      In quantum mechanics, a quantum operation is a morphism in the category of density matrices

      at the beginning of the Idea section motivated by O’Loan’s comment

      A quantum channel is a mapping which sends density matrices to density matrices.

      This seems innocent enough, but someone might check the statement. For one, I’ve never seen a category of density matrices, but the idea seems obvious enough. Maybe a word on density matrix would be good.

    • Hello,

      I noticed DFT page has not been updated in a while and I added a couple of sections: some sketchy introductory material (analogy between Kaluza-Klein and DFT) and a little insight about a more rigorous geometrical formulation of DFT.

      It is still quite sketchy but I would be happy to refine it.

      PS: this is my first edit, I hope I played by the rules. And thank you all for this wiki

      Luigi

      diff, v7, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • changed “involution” to “anti-involution”

      diff, v61, current

    • Created a stub for the conference.

      v1, current

    • added pointer to today’s

      • Andrea Fontanella, Tomas Ortin, On the supersymmetric solutions of the Heterotic Superstring effective action (arxiv:1910.08496)

      diff, v57, current

    • have added some minimum of references (there were none before)

      but I hope to find the time to put some actual content into the entry:

      the sequence of exceptional tangent bundles used to be truncated, and the other day I saw (cf. nForum discussion here and here) how to complete it, using recent results.

      a pdf note is now here (just 1 page)

      diff, v5, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I’ve created a page for quaternionic manifolds, linked from quaternion-Kähler page. Basic references and discussion of main definition re Cauchy-Feuter calculus. Comparison to hypercomplex structure.

      v1, current

    • 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!

    • This is for olog-specific stuff which wouldn’t be appropriate for biology.

      v1, current

    • Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Added redirect for “E9” at the bottom of the page.)

      diff, v7, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I have half-heartedly started adding something to Kac-Moody algebra. Mostly refrences so far. But I don’t have the time right now to do any more.

    • have added a minimum on the level decompositon of the first fundamental rep of E 11E_{11} here.

    • created traced monoidal category with a bare minimum

      I would have sworn that we already had an entry on that, but it seems we didn’t. If I somehow missed it , let me know and we need to fix things then.

    • Added:

      The large cardinal strength of the weak Vopěnka principle is discussed in

      The following paper shows that weak Vopěnka’s principle is indeed weaker than Vopěnka’s principle:

      diff, v21, current

    • I am touching various entries related to equivariant stable homotopy theory, adding basics from the literature. For instance I briefly added to G-spectrum the basic definition via indexing on a universe, and added the statement of the equivariant stable Whitehead theorem, cross-linked with the relevant bits at equivariant homotopy theory, etc. I have also been expanding a little more at RO(G)-grading and cross-linked more with old material at equivariant cohomology. Tried to make the link between RO(G)-grading and equivariant suspension isomorphism more explicit.

      Just in case you are watching the logs and are wondering. I am not announcing every single edit, unless there is anything noteworthy.

    • This entry used to be named “crystal”. Re-named hereby to make room for a page on the actual notion in solid state physics. Will add disambiguation redirects.

      diff, v17, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • started some minimum at exceptional field theory (the formulation of 11d supergravity that makes the exceptional U-duality symmetry manifest)

    • brief category:people-entry for hyperlinking references

      v1, current

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