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I have renamed the entry on the -topos on 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 sends paracompact topological spaces to their traditional fundamental -groupoid
;
more generally, for a simplicial topological space we have
,
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 DO Bundles (arXiv:math/0301185)
Steven Rosenberg, Fabián Torres-Ardila, Infinite Dimensional Chern-Simons Theory (arXiv:math/0411161)
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
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 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 is. So it seems clearly wrong that you can't make 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 . 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 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 will (as long as isn’t already compact) have more points than ; but by the isomorphism , points of an exponential space have to be in bijection with continuous maps .
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 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 as the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object from
moving material about the weak limited principle of omniscience from principle of omniscience to its own page at weak limited principle of omniscience
Anonymouse
After a suggestion from Toby, I added a note on the “analytic Markov’s principle” to Markov’s principle.
moving material about the lesser limited principle of omniscience from principle of omniscience to its own page at lesser limited principle of omniscience
Anonymouse
added to homotopy groups of spheres the table
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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?)
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.)
created black holes in string theory, since somebody asked me: a brief paragraph explaining how the entropy-counting works and some references.
Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.
Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.
Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.
Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.
I would try to say what the idea is, but have to dash.
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. 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 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 ]
I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.
Changes made only to the Universal property of the 2-category of spans section. The citations by Urs lead to another citation which, in turn, leads to another citation. With a little effort, I tracked down the a full copy of said universal property, I’ve replicated it here, added the citation used, although I left the previous citations there for convenience; a more experienced editor can remove those if they would like.
I would like to note that the author whose work I have referenced, Hermida, also notes: “[this universal property] is folklore although we know no references for it.”
Please make any corrections needed and clean up the language here; this is a fairly direct copy of what is written, but I imagine somebody with more knowledge of all the language used here can rewrite this universal property stuff in a cleaner way.
Thanks!
Anonymous
stub for modular functor
starting a dedicated entry for the category of vector bundles with homomorphisms allowed to cover non-trivial base maps (while previously we only had VectBund(B) for fixed base ).
For the moment the main point is to record the interesting cartesian- and tensor-monoidal structure (now here)
I will brush-up the entry homotopy hypothesis. But not right now, right now I have to run and do something else. But here is some leftover discussion that was sitting there, and which I have now removed from the entry and reproduce here, in order that we go and use it to make the entry better, but not clutter it up.
(continued in next comment)
Copied some writings from other articles, added related entries and added Wikipedia entry.
I made “constructive logic” redirect to here (“constructive mathematics”) instead of to “intuitionistic mathematics”, as it used to
At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group together with the bundary map and the -action on . As someone confirms this example is correct I’ll add it to crossed module.
added pointer to:
brief category:people
-entry for hyperlinking references at string field theory
added pointer to Elliott-Safronov 18