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Expansion of references section at differential topology.
Explained why this definition is equivalent to the one at van Kampen colimit.
a bare list of references, to be !include-ed into relevant entries, such as at elliptic cohomology, but also at equivariant elliptic cohomology, elliptic genus, Witten genus etc.
(in an attempt to clean up and harmonize the referencing across all these entries – still some way to go towards that goal, but it should be a start)
have added a minimum on the level decompositon of the first fundamental rep of E11 here.
There was a section about W-suspensions titled “Higher inductive types generated by graphs” in the article coequalizer type, so I moved the section into its own page at W-suspension.
Created this article on graph quotients using material split from W-suspension. The “W-suspensions in the first sense” in that article are actually called graph quotients in
I have added to continuum a paragraph titled In cohesive homotopy type theory.
This is a simple observation and idea that I have been carrying around for a while. Several people are currently thinking about ways to axiomatize the reals in (homotopy) type theory.
With cohesive homotopy type theory there is what looks like an interesting option for an approach different to the other ones: one can ask more generally about line objects 𝔸1 that look like continua.
One simple way to axiomatize this would be to say:
𝔸 is a ring object;
it is geometrically contractible, Π𝔸≃*.
The last condition reflects the “continuumness”. For instance in the standard model Smooth∞Grpd for smooth homotopy cohesion, this distibuishes 𝔸=ℤ,ℚ from 𝔸=ℝ,ℂ.
So while this axiomatization clearly captures one aspect of “continuum” very elegantly, I don’t know yet how far one can carry this in order to actually derive statements that one would want to make, say, about the real numbers.
just heard an interesting talk by Steven Rosenberg on CS invariants on infinite-dimensional manifolds. So I created an entry infinite-dimensional Chern-Simons theory in order to record some references
Linked new person entries for Eleny-Nicoleta Ionel and Thomas H. Parker.
added pointer to:
Added content, including the idea, GW/PT/GV correspondence and references. (The german Wikipedia article is now also available.)
(I also plan to create an article for the Pandharipande-Thomas invariant in the future.)
felt like adding a handful of basic properties to epimorphism
felt the desire to have an entry on the general idea (if any) of synthetic mathematics, cross-linking with the relevant examples-entries.
This has much room for being further expanded, of course.
I have added to M5-brane a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).
In the new section Conformal blocks and 7d Chern-Simons dual.
The following sentence was at inaccessible cardinal:
A weakly inaccessible cardinal may be strengthened to produce a (generally larger) strongly inaccessible cardinal.
I have removed it after complaints by a set theorist, and in light of the below discussion box, which I have copied here and removed.
Mike: What does that last sentence mean? It seems obviously false to me in the absence of CH.
Toby: It means that if a weakly inaccessible cardinal exists, then a strongly inaccessible cardinal exists, but I couldn't find the formula for it. Something like ℶκ is strongly inaccessible if κ is weakly inaccessible (note that ℵκ=κ then), but I couldn't verify that (or check how it holds up in the absence of choice).
Mike: I don’t believe that. Suppose that the smallest weakly inaccessible is not strongly inaccessible, and let κ be the smallest strongly inaccessible. Then Vκ is a model of set theory in which there are weakly inaccessibles but not strong ones. I’m almost certain there is no reason for the smallest weakly inaccessible to be strongly inaccessible.
JCMcKeown: Surely ℶκ has cofinality at most κ, so it can’t be regular. Maybe the strengthening involves some forcing or other change of universe? E.g., you can forcibly shift 2λ=λ+ for λ<κ, and then by weak inaccessibility, etc… I think. Don’t trust me. —- (some days later) More than that: since the ordinals are well ordered, if there is any strongly inaccessible cardinal greater than κ, then there is a least one, say θ. Then Vθ is a universe with a weakly inaccessible cardinal and no greater strongly inaccessible cardinal. Ih! Mike said that already… So whatever construction will have to work the other way around: if there is a weakly inaccessible cardinal that isn’t strongly inaccessible, and if furthermore a weakly inaccessible cardinal implies a strongly inaccessible cardinal, then the strongly inaccessible cardinal implied must be less than κ. And that sounds really weird.
I was also supplied with a AC-free proof that weakly inaccessibles are ℵ(−)-fixed points:
First, a quick induction shows that α≤ωα is always true. If κ is a limit cardinal, then the set of cardinals below κ is unbounded; but since it’s also regular there are κ of them. So ωκ≤κ≤ωκ, and equality ensues.
I can edit this into the page if desired.
I have renamed the entry on the ∞-topos on CartSptop into Euclidean-topological infinity-groupoid.
Then in the section Geometric homotopy I have written out statement and proof that
the intrinsic fundamental ∞-groupoid functor in ETop∞Grpd sends paracompact topological spaces to their traditional fundamental ∞-groupoid
ΠETop∞Grpd(X)≃ΠTop(X)≃SingX;
more generally, for X• a simplicial topological space we have
|ΠETop∞Grpd(X•)|≃|X•| ,
where on the left we hve geometric realization of simplicial sets, and on the right of (good) simplicial topological spaces.
Copied from infinite-dimensional Chern-Simons theory:
Sylvie Paycha, Steven Rosenberg, Chern-Weil Constructions on ΨDO Bundles (arXiv:math/0301185)
Steven Rosenberg, Fabián Torres-Ardila, Infinite Dimensional Chern-Simons Theory (arXiv:math/0411161)
Started the article dependent choice, and did some editing at COSHEP to make clearer to myself the argument that COSHEP + (1 is projective) implies dependent choice. It’s not clear to me that the projectivity of 1 is removable in that argument; maybe it is.
moving the following old discussion from out of the entry to here, just for the record (it concerns a bygone version of the entry):
+– {: .query} Tim: As I read the entry on nice topological spaces, it really refers to ’nice categories’ rather than ’nice spaces’! I have always thought of spaces such as CW-complexes and polyhedra as being ’locally nice’, but the corresponding categories are certainly not ’nice’ in the sense of nice topological space. Perhaps we need to adjust that other entry in some way.
Toby: You're right, I think I've been linking that page wrongly. (I just now did it again on homotopy type!) Perhaps we should write locally nice space or locally nice topological space (you pick), and I'll fix all of the links tomorrow.
Tim:I suggest locally nice space. (For some time I worked in Shape Theory where local singularities were allowed so the spaces were not locally nice!) There would need to be an entry on locally nice. I suggets various meanings are discussed briefly, e.g. locally contractible, locally Euclidean, … and so on, but each with a minimum on it as the real stuff is in CW-complex etc and these are the ’ideas’.
Mike: Why not change the page nice topological space to be about CW-complexes and so on, and move the existing material there to something like convenient category of spaces, which is also a historically valid term? I am probably to blame for the current misleading content of nice topological space and I’d be happy to have this changed.
Toby: I thought that nice topological space was supposed to be about special kinds of spaces, such as locally compact Hausdorff spaces, whose full subcategories of Sp are also nice. (Sort of a counterpoint to the dichotomy between nice objects and nice categories, whose theme is better fit by the example of locally Euclidean spaces). CW-complexes also apply —if you're interested in the homotopy categories.
Mike: Well, that’s not what I thought. (-: I don’t really know any type of space that is nice and whose corresponding subcategory of Top is also nice. The category of locally compact Hausdorff spaces, for instance, is not really all that nice. In fact, I can’t think of anything particularly good about it. I don’t even see any reason for it to be complete or cocomplete!
I think it would be better, and less confusing, to have separate pages for “nice spaces” and “nice categories of spaces,” or whatever we call them. And, as I said, I don’t see any need to invent a new term like “locally nice.”
When algebraic topologists (and, by extension, people talking about ∞-groupoids) say “nice space” they usually mean either (1) an object of some convenient category of spaces, or (2) a CW-complex-like space, between which weak homotopy equivalences are homotopy equivalences. Actually, there is a precise term for the latter sort: an m-cofibrant space, aka a space of the (non-weak) homotopy type of a CW complex.
Toby: I thought the full subcategory of locally compact Hausdorff spaces was cartesian closed? Maybe not, and it's not mentioned above.
But you can see that most of the examples above list nice properties of their full subcategories. And the page begins by talking about what a lousy category Top is. So it seems clearly wrong that you can't make Top a nicer category by taking a full subcategory of nice spaces. (Not all of the examples are subcategories, of course.)
Mike: It’s true that locally compact Hausdorff spaces are exponentiable in Top. However, I don’t think there’s any reason why the exponential should again be locally compact Hausdorff.
I guess you are right that one could argue that compactly generated spaces themselves are “nice,” although I think the main reason they are important is that the category of compactly generated spaces is nice. I propose the following:
Toby: I believe that the compact Hausdorff reflection (the Stone–Čech compactification) of YX is an exponential object.
Anyway, your plan sounds fine, although nice category of spaces might be another title. (I guess that it's up to whoever gets around to writing it first.) Although I'm not sure that people really mean m-cofibrant spaces when they speak of nice topological spaces when doing homotopy theory; how do we know that they aren't referring to CW-complexes? (which is what I always assumed that I meant).
Mike: I guess nice category of spaces would fit better with the existing cumbersomely-named dichotomy between nice objects and nice categories. I should have said that when people say “nice topological space” as a means of not having to worry about weak homotopy equivalences, they might as well mean (or maybe even “should” mean) m-cofibrant space. If people do mean CW-complex for some more precise reason (such as wanting to induct up the cells), then they can say “CW complex” instead.
Re: exponentials, the Stone-Čech compactification of YX will (as long as YX isn’t already compact) have more points than YX; but by the isomorphism Hom(1,YX)≅Hom(X,Y), points of an exponential space have to be in bijection with continuous maps X→Y.
Toby: OK, I'll have to check how exactly they use the category of locally compact Hausdorff spaces. (One way is to get compactly generated spaces, of course, but I thought that there was more to it than that.) But anyway, I'm happy with your plan and will help you carry it out.
=–
I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for Set and made some remarks on exponentiation of numbers.
For some time now I’ve been bothered by an implicit redundancy spanned by the articles nice category of spaces and convenient category of topological spaces. I would like the latter to have a more precise meaning and the former to be something more vague and flexible. I have therefore been doing some rewriting at the former. But if anyone disagrees with the edits, please let’s discuss this here.
I have removed a query box:
+– {: .query} I’m not sure that we really want to use the terminology that way, but Ronnie already created that page, so I’m linking these together. —Toby =–
added pointer to
adding reference
Maus
I have updated this item:
with the arXiv number. It looks like a bunch of recent anonymously-authored entries are essentially taken from this thesis (such as protoring and its many variants).
The AnonymousCoward who creates blank pages in places where I ought to write stubs has been at it again, this time at Stone duality.
Added the characterization of Comp as the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object from
added to homotopy groups of spheres the table
| k= | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ⋯ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| πk(𝕊)= | ℤ | ℤ2 | ℤ2 | ℤ24 | 0 | 0 | ℤ2 | ℤ240 | (ℤ2)2 | (ℤ2)3 | ℤ6 | ℤ504 | 0 | ℤ3 | (ℤ2)2 | ℤ480⊕ℤ2 |
Added:
Rajesh Gopakumar, Cumrun Vafa, M-Theory and Topological Strings–I (1998), (arXiv:hep-th/9809187, bibcode:1998hep.th….9187G)
Rajesh Gopakumar, Cumrun Vafa, M-Theory and Topological Strings–II (1998), (arXiv:hep-th/9812127, bibcode:1998hep.th…12127G)
(On the Gauge Theory/Geometry Correspondence is in there twice. Is that supposed to be?)
the page action is also a mess. I have added a pointer to the somewhat more comprehensive module and am hereby moving the following discussion box from there to here:
[ begin forwarded discussion ]
+–{.query} I am wondering if we will need the notion of action which works in categories with product, i.e. G×X→X and so on. There is also an action of one Lie algebra on another (for instance in some definitions of crossed module of Lie algebra, where Aut is replaced by the Lie algebra of derivations. (a similar situation would seem to exist in various other categories where action is needed in a slightly wider context. I think most would be covered by an enriched setting but I am not sure.) Thoughts please.Tim
Yes, I think certainly all those types of action should eventually be described somewhere, possibly on this page. -Mike
Tim: I have added some of this above. There should be mention of actions of a monoid in a monoidal category on other objects, perhaps.
Mac Lane, VII.4, only requires a monoidal category to define actions. – Uday =–
[ end forwarded discussion ]