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a stub entry, to give a home to today’s
stub for Poisson sigma-model. Needs references.
I have created an entry on the quaternionic Hopf fibration and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in G-equivariant stable homotopy theory it represents a non-torsion element in
[Σ∞GS7,Σ∞GS4]G≃ℤ⊕⋯for G a finite and non-cyclic subgroup of SO(3), and SO(3) acting on the quaternionic Hopf fibration via automorphisms of the quaternions.
I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.
Added the reference:
Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).
Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.
started Euler class
For completeness, so that we now have this list:
the brief idea at kinematics and dynamics
Not an edit, but is there anything concrete known about this kind of automorphism group for an infinity-group? Say its homotopy type?
Redirect projective group, as it is usually called in the context of projective geometry (classical at least).
added pointer to:
Adding reference
Anonymouse
Moved the definition of constant functor from cone to a new page constant functor.
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.
=–
Starting page on the “principle of enough functions”: the formal locale of the function space ℝℝ of continuous real endofunctions is a spatial locale, as described in
Anonymouse
stub for Heine-Borel theorem
Added reference
note that the website linked on this page doesn’t work anymore
Anonymouse
category: people page for reference
Anonymouse
Change 1: Original page describes the fan theorem as requiring the bar to be decidable, claims that the “classical” fan theorem contradicts Brouwer’s continuity principle. The latter claim is not true; I corrected the error. I have stated the result as two separate theorems: the decidable fan theorem, about decidable bars, and the fan theorem, about bars in general.
Change 2: Slightly more information is provided about the relationship between the Fan Theorem and Bar Induction. Eventually, we should make a page about the latter.
Change 3: the section on equivalents to the fan theorem has been fixed somewhat. The section originally asserted that all of the statements provided were equivalent to the decidable fan theorem; in fact, some are equivalent to the decidable fan theorem and some to the full fan theorem.
As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)
category: people page for references
Tatsuji Kawai, Giovanni Sambin, The principle of pointfree continuity, Logical Methods in Computer Science, Volume 15, Issue 1 (March 5, 2019). (doi:10.23638/LMCS-15%281%3A22%292019, arXiv:1802.04512)
Tatsuji Kawai, Principles of bar induction and continuity on Baire space (arXiv:1808.04082)
Tatsuji Kawai, Representing definable functions of HAω by neighbourhood functions (arXiv:1901.11270)
Anonymouse
Added reference
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category: people page for reference
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category: people page for reference
found in bar induction
Anonymouse
added some content to variational bicomplex
briefly added something to fusion category. See also this blog comment.
stub for Stone-von Neumann theorem
Added a budget of links from the FAQ to the Eunuch-Code Data-Bank.
For some odd reason, the TOC generator is not picking up the first heading — ???
I added some simpler motivation in terms of the basic example to the beginning of distributive law.
Added the first paper on event structures:
gave representation theory a little Idea-section, then added some words on its incarnation as homotopy type theory in context/in the slice over BG and added the following homotopy type representation theory – table, which I am also including in other relevant entries:
| homotopy type theory | representation theory |
|---|---|
| pointed connected context BG | ∞-group G |
| dependent type | ∞-action/∞-representation |
| dependent sum along BG→* | coinvariants/homotopy quotient |
| context extension along BG→* | trivial representation |
| dependent product along BG→* | homotopy invariants/∞-group cohomology |
| dependent sum along BG→BH | induced representation |
| context extension along BG→BH | |
| dependent product along BG→BH | coinduced representation |