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I wrote about Dmitri Pavlov’s concept of measurable locales.
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)
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?
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)
brief category:people-entry for hyperlinking references at Randall-Sundrum model
for hyperlinking references at Randall-Sundrum model and at probe brane
brief category:people-entry for hyperlinking references at flavor brane and at probe brane
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 (or as is now more common) for a hom-set. But personally, I like (or if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from to . —Toby Bartels
Mike: Perhaps. I never write for a hom-set, only or where is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen 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) or if you are considering it as a discrete fibration from to . 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 as less visually distracting, and evidently a generalization of the common notation for a slice category.
Toby: Well, I never stick ‘’ 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 for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see in a category is that we have and and we’re talking about the pair — surely also a natural generalization of the very well-established notation for ordered pairs.
Toby: The notation for a double comma object makes me like even more!
Mike: I’d rather avoid using in the name of an object; talking about projections looks a good deal more confusing to me than .
Toby: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If , then ought to be the set of transformations between them. (Or , 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 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 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 and 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 to , that are hom-sets. Finally, I don’t think the notation scales well to double comma objects; we could write 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 . Maybe ? Lengthy, but at least unambiguous. Or maybe ?
Zoran Skoda: or are the only two standard notations nowdays, I think the original which was done for typographical reasons in archaic period is abandonded by the LaTeX era. is more popular among practical mathematicians, and special cases, like when ) and among category experts…other possibilities for notation should be avoided I think.
Urs: sounds good. I’ll try to stick to then.
Mike: There are many category theorists who write , including (in my experience) most Australians. I prefer myself, although I occasionally write if I’m talking to someone who I worry might be confused by .
Urs: recently in a talk when an over-category appeared as somebody in the audience asked: “What’s that quotient?”. But already looks different. And of course the proper even more so.
Anyway, that just to say: i like , find it less cumbersome than and apologize for having written so often.
Toby: I find more self explanatory, but is cool. 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.
Added a description of several different approaches to geometric theory.