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    • Created page, copying material from the one on David Roberts’ web.

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • Added a reference.

      Can we say exactly what kind of pretopos the category of small presheaves on a category C is?

      Is it a ΠW-pretopos, provided that PC is complete?

      diff, v9, current

    • a bare sub-section with a list of references – to be !included into relevant entries – mainly at confinement and at mass gap problem (where this list already used to live)

      v1, current

    • some minimum, just so that I can link to it

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    • starting some minimum, for the moment mostly to have a place for collecting references

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    • I added a reference to an early talk on Yoneda structures in which Walters advertizes them as a 2-dimensional version of the category of categories.

      diff, v30, current

    • Page created, but author did not leave any comments.

      v1, current

    • Started this having heard someone mention it.

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    • a stub, just so that the link works

      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 (fg)(f \rightarrow g) (or (fg)(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:XAf:X\to A and g:XBg:X\to B and we’re talking about the pair (f,g):XA×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 (fgh)(f \to g \to h) even more!

      Mike: I’d rather avoid using \to in the name of an object; talking about projections p:(fg)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:ABf, g: A \to B, then fgf \to g ought to be the set of transformations between them. (Or fgf \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[CfEgD]M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]. Maybe comma[CfEgD]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 (fg)(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 (fg)(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 (fg)(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 (fg)(f\downarrow g) and apologize for having written (f,g)(f,g) so often.

      Toby: I find (fg)(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.

      =–

    • stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.

    • fixed spelling of the word constraint (one instance) in paragraph under ### Gravity as a (non-)gauge theory

      cofo

      diff, v56, current

    • Add missing separation condition that Stone uses in his proof. Explain why it’s necessary.

      diff, v3, current

    • Fixed a link to point to his departmental ’profile’.

      diff, v4, current

    • Mention that Hausdorff is not required in the first Michael theorem (almost all standard sources, including Michael himself, impose it).

      diff, v10, current