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I moved most of the contents of the material from preconvergence space to a different article, since “preconvergence space” is evidently defined in the existing literature as a different thing than what the original article says.
I also added a disclaimer at the top of the page that the name of the article is just a placeholder name.
Created page for BO(n), the classifying space of the orthogonal group O(n). (See discussion on Stiefel-Whitney class.) There is still a lot to add though.
I’m not entirely happy with the introduction (“Statement”) to the page axiom of choice. On the one hand, it implies that the axiom of choice is something to be considered relative to a given category C (which is reasonable), but it then proceeds to give the external formulation of AC for such a C, which I think is usually not the best meaning of “AC relative to C”. I would prefer to give the Statement as “every surjection in the category of sets splits” and then discuss later that analogous statements for other categories (including both internal and external ones) can also be called “axioms of choice” — but with emphasis on the internal ones, since they are what correspond to the original axiom of choice (for sets) in the internal logic.
(I would also prefer to change “epimorphism” for “surjection” or “regular/effective epimorphism”, especially when generalizing away from sets.)
Added a literature reference to icon. Started some systematic notes on icons for monoidal-enriched bicategories, which I am currently using for something. Think the broken-off state of that section is not intolerable, in particular since I have seen similar work in progress on the nLab. Intend to continue them soon.
I have added the version of the axioms for reduced cohomology here at generalized (Eilenberg-Steenrod) cohomology (and also at reduced cohomology), and I have further expanded and streamlined (I hope) the Idea-section.
More harmonization (notation, conventions) is necessary in this and related entries. Later.
Noticing that the term “gauge field” used to conflictingly be redirecting both to “gauge theory” and to “field (physics)”, neither of which is satisfactory as a redirect, I am giving the term its own entry hereby.
But it’s just a stub entry for now.
created Frobenius monoidal functor
am starting power operation, but nothing there yet except references
Here is old discussion that used to be in the entry graph and which hereby I am moving to the relevant talk-page (i.e.: the nForum thread with the same title as the entry, namely this one).
[begin forwarded discussion]
Obsolete discussion may also be found in the History at Version 24.
Toby: OK, I've completely redone the page above; this is how it looked before. In particular, I am defining things case by case, rather than choice by choice (8 cases, rather than 3 choices with 2 options each). Feedback please!
(One obvious possibility is that the best style of definition is a mixture of the two previous styles: doing undirected and directed graphs separately, but in each case listing the two choices —loops or no loops, multiple edges or no multiple edges— as I had done before.)
Eric: Ugh. I see that quite some discussion went on here and I’m late to the party. This page is not beautiful nor remotely n-categorical in my opinion. We already had a page that I was very happy with on directed graphs.
Isn’t there some way to state very simply:
A graph is a functor…
Here is a humble attempt…
+– {: .un_defn}
An abstract graph X is a category with
one object X0, called the object of vertices;
one object X1, called the object of edges;
one morphism e:X1→???, called the ???;
together with identity morphisms.
A graph is a functor G:X→ Set.
More generally, a graph in a category C is a functor G:X→C. =–
Toby: First, it depends on what kind of ’graph’ you mean.
Let's take a simple undirected graph. Then the answer is no, since the definition of a simple graph is not (despite the name) as simple as the definition of digraph (directed pseudograph). Whereas a digraph consists of just V, E, and d:E→V2, a simple graph consists of V, E, and an injection d:E→(V2). The two problems here are: how do you say that d is an injection? and how do you describe a function E→(V2) in terms of functions among V and E? (A map E→V2 can be done; that's the same as two maps E→V.) You can describe these things more internally, of course (say by replacing ’injective function’ with ’monomorphism’), but there's no category X such that a simple graph is precisely a functor from X to Set.
In fact, the only kind of graph above that can be defined as a functor from X to Set for some fixed ’abstract general’ category X is directed pseudograph, the kind of graph discussed at digraph. Between that, and the fact that every strict category has an underlying digraph, it's no surprise that this is the sort of graph that category theorists like. But it's not the sort of graph that graph theorists like so much!
It would be worth discussing what sort of graphs can be internalised in what sort of categories. Those graphs that allow loops are easier; I think that I can do them! For the graphs without loops, I haven't even decided what's the best way to phrase the definition in constructive mathematics. (Luckily it doesn't matter for finite graphs.)
[forwarded discussion continued in next comment]
created induced metric, just for completeness
added more references:
Paolo Salvatore: Configuration spaces with summable labels, in Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math. 196, Birkhäuser (2001) 375–395 [doi:10.1007/978-3-0348-8312-2_23, arXiv:math/9907073]
Jeremy Miller: Nonabelian Poincaré duality after stabilizing, Trans. Amer. Math. Soc. 367 (2015) 1969-1991 [doi:2015-367-03/S0002-9947-2014-06186-2, arXiv:1209.2773]
Sadok Kallel: Particle Spaces on Manifolds and Generalized Poincaré Dualities, Quarterly J. Math. 52 1 (2001) 45–70 [doi:10.1093/qjmath/52.1.45, arXiv:math/9810067]
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) (or [x,y] as is now more common) for a hom-set. But personally, I like (f→g) (or (f↘g) if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from f to g. —Toby Bartels
Mike: Perhaps. I never write (x,y) for a hom-set, only A(x,y) or homA(x,y) where A is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen [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) or homE(f,g) if you are considering it as a discrete fibration from A to B. 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) as less visually distracting, and evidently a generalization of the common notation C/x for a slice category.
Toby: Well, I never stick ‘E’ 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) for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see (f,g) in a category is that we have f:X→A and g:X→B and we’re talking about the pair (f,g):X→A×B — surely also a natural generalization of the very well-established notation for ordered pairs.
Toby: The notation (f/g/h) for a double comma object makes me like (f→g→h) even more!
Mike: I’d rather avoid using → in the name of an object; talking about projections p:(f→g)→A looks a good deal more confusing to me than p:(f/g)→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→B, then f→g ought to be the set of transformations between them. (Or f⇒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) 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) 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 f and g 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 C to D, that are hom-sets. Finally, I don’t think the notation (f,g) scales well to double comma objects; we could write (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[Cf→Eg←D]. Maybe comma[Cf→Eg←D]? Lengthy, but at least unambiguous. Or maybe fEIg?
Zoran Skoda: (f/g) or (f↓g) are the only two standard notations nowdays, I think the original (f,g) which was done for typographical reasons in archaic period is abandonded by the LaTeX era. (f/g) is more popular among practical mathematicians, and special cases, like when g=idD) and (f↓g) among category experts…other possibilities for notation should be avoided I think.
Urs: sounds good. I’ll try to stick to (f/g) then.
Mike: There are many category theorists who write (f/g), including (in my experience) most Australians. I prefer (f/g) myself, although I occasionally write (f↓g) if I’m talking to someone who I worry might be confused by (f/g).
Urs: recently in a talk when an over-category appeared as C/a somebody in the audience asked: “What’s that quotient?”. But (C/a) already looks different. And of course the proper (IdC/consta) even more so.
Anyway, that just to say: i like (f/g), find it less cumbersome than (f↓g) and apologize for having written (f,g) so often.
Toby: I find (f↓g) more self explanatory, but (f/g) is cool. (f,g) was reasonable, but we now have better options.
=–
Have added DOI-s to these:
Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850 (doi:10.1007/s002220100175, pdf)
Matthew Ando, Michael Hopkins, Neil Strickland, The sigma orientation is an H-infinity map, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (arXiv:math/0204053, doi:10.1353/ajm.2004.0008)
So this one here remains unpublished:
?
since this was missing, I created a minimum at equivalence in a 2-category
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
motivated by the blog discussion I added to rational homotopy theory a section Differential forms on topological spaces
A stub, for the moment just to have a place for recording a couple of references (which were previously at fusion category.
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.
Created apartness space.
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.
A stub, to make links work at Wheeler superspace
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.
added pointer to today’s
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)
Created reflexive coequalizer.
wrote an Idea-section at quantum field theory
created hyperring
stub (except for a brief remark on this being the gauge group in type I ST), for the moment just to fill the pattern at low dimensional rotation groups – table
am finally adding references here, such as
will add these also to lattice gauge theory as far as there is overlap
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.)
a bare list of references, to be !include-ed into the References-lists of relevant entries (inductive type, inductive familiy, inductive-recursive family, calculus of inductive constructions)
this list includes a polished-up version of
all the references previously listed at inductive familiy in the section “History” (due to revisions ≤5 by Bas Spitters)
further references previously listed at inductive-recursive type
and some more
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:
Created new article for Bi-Yang-Mills equation. (The english and german Wikipedia article are now also available.)