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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have split off complex projective space from projective space and added some basic facts about its cohomology.
Added doi and pointer to relevant sections to
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 6 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf, doi:10.1007/b97586)
(EM-spaces are constructed in section 6, the cohomology theory they represent is discussed in section 7.1, and its equivalence to singular cohomology is Corollary 12.1.20)
following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be !include-ed into the References sections of relevant entries (mainly at homotopy theory and at algebraic topology) for ease of updating and syncing these lists.
The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.
I’ll just check now that I have all items copied, and then I will !include this entry here into homotopy theory and algebraic topology. It may best be viewed withing these entries, because there – but not here – will there be a table of contents showing the subsections here.
wrote Maurer-Cartan form
the first part is the standard story, but I chose a presentation which I find more insightful than the standard symbol chains as on Wikipedia.
then there is a section on Maurer-Cartan forms on oo-Lie groups and how that reduces to the standard story for ordinary Lie groups.
The detailed statements and proofs of this second part are at Lie infinity-groupoid in the new section The canonical form on a Lie oo-group that is just a Lie group.
added to convenient vector space a Properties-section mentioning their embedding into the Cahiers topos, and added the reference by Kock where this is proven.
a bare list of references, to be !include-ed into the References-sections of relevant entries, for ease of synchronization (at exceptional field theory, exceptional geometry, sigma-model and maybe super p-brane)
Added a reference to the following which provides a proof of the Arnold conjecture
added to exceptional generalized geometry two examples of reductions of stucture groups that encode higher supersymmetry in 11d sugra.
added to tmf a section that gives an outline of the proof strategy for how to compute the homotopy groups of the -spectrum from global sections of the -structure sheaf on the moduli stack of elliptic curves.
A point which I wanted to emphasize is that
The problem of constructing as global sections of an -structure sheaf has a tautological solution: take the underlying space to be .
From this tautological but useless solution one gets to the one that is used for actual computations by one single crucial fact:
In the -topos over the -site of formal duals of -rings, the dual of the Thom spectrum, is a well-supported object. the terminal morphism
in the -topos is an effective epimorphism, hence a covering of the point.
Using this we can pull back the tautological solution of the problem to the cover and then compute there. This is what actually happens in practice: the decategorification of the pullback of to is the moduli stack of elliptic curves. And it is a happy coincidence that despite this drastic decategorification, there is still enough information left to compute on that.
added pointer to today’s
Added to Hopf monad the Bruguières-Lack-Virelizier definition and some properties.
I split the section on cluster spaces in the convergence space article off to its own article.
creating the counterpart to preconvergence space but for sequences only instead of for all nets.
Anonymous
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 (which is reasonable), but it then proceeds to give the external formulation of AC for such a , which I think is usually not the best meaning of “AC relative to ”. 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
brief category:people-entry for satisfying links now requested at p-adic Teichmüller theory
I have expanded vertex operator algebra (more references, more items in the Properties-section) in partial support to a TP.SE answer that I posted here
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 Forum 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 ( cases, rather than choices with 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 -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 is a category with
one object , called the object of vertices;
one object , called the object of edges;
one morphism , called the ???;
together with identity morphisms.
More generally, a graph in a category is a functor . =–
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 , , and , a simple graph consists of , , and an injection . The two problems here are: how do you say that is an injection? and how do you describe a function in terms of functions among and ? (A map can be done; that's the same as two maps .) You can describe these things more internally, of course (say by replacing ’injective function’ with ’monomorphism’), but there's no category such that a simple graph is precisely a functor from to .
In fact, the only kind of graph above that can be defined as a functor from to for some fixed ’abstract general’ category 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]
started a stub for ambidextrous adjunction, but not much there yet
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 (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.
=–
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