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At topological vector space, there's a spot where a uniform space is characterised by giving a base of entourages. Zoran thought that it would be a good way to make clear that ‘entourage’ is a technical term by making it into a link. So now there is a page entourage.
Most of the details are still to be found at uniform space, however.
I am working on the next chapter in geometry of physics:
This is not done yet, but it should already be readable. To some extent I am taking the talk notes Super Lie n-algebra of Super p-branes (schreiber) and expand them into fully fledged lecture notes.
Since I am editing in a separate window, at this point I ask that everyone who feels like touching this page, even if it just concerns tiny changes (typos) to please alert me.
I am working on giving the entry on topology a section Introduction. This section is meant to provide persons with some background in, say, analysis, but otherwise with no idea of topology, briefly with some basic ideas. The basic definition, some pictures, the basic idea of how to use topological invariants in very simple examples, maybe culminating in an outline of the fundamental group and its relation to covering spaces. Not done yet.
I put the constructive version of coalgebra of the real interval there and straightened up the references.
Suppose somebody formally minded is looking for a good problem in the topic of contextual categories (C-systems). How about this:
fix a given C-system, with your preferred set of extra type constructor data on it, and then ask the question: for any given small site, is the category of sheaves on that site with values in that C-system again canonically a C-system with the same collection of extra type constructors?
I gather aspects of this play a role in most discussions of type theory model building, but is there any systematic discussion?
I suppose the difficulty and interest in this question considerably varies with what the set of “extra type constructors” includes. A while back I had asked a similar question where “extra type constructors” was “modalities”. Maybe that was overambitious for the person who I am asking for, so I am trying to see if something along these lines but more tractable would be a good thing to aim at.
The entry closed subspace was a bit weird. I have touched it to try to improve a little. But if anyone has ten minutes to spare, it might still be good to bring this into more decent shape.
To make sense of this in my mind as a general concept, I have written semidefinite element. This gives a general context in which to define ‘positive definite’, ‘negative semidefinite’, ‘indefinite’ etc.
(This seems the safest page title, as the least likely to have any conflicts. I've also put in a lot of redirects, but possibly some of these will have to go elsewhere; definite seems the most dangerous.)
quick entry Haag–Lopuszanski–Sohnius theorem
contractible object redirected to both sufficiently cohesive topos and contractible space. Since neither or these really seemed to fit, I removed both of them. (And then I put a link to contractible object in sufficiently cohesive topos, but contractible space doesn't have a place for one.)
By the way, how long has it been true that if you follow a link that has more than one redirect that you get a warning about this?
started stubs for ring of adeles and group of ideles – but they are not good yet
Somehow I was under the impression that I had written out on the Lab at several places how the traditional physics way to talk about instantons connects to the correct maths discussion. But now that I wanted to point a physicist to this, I realize that in each entry that touches on this, I just gave a quick remark pointing to Cech cohomology, clutching construction, one-point compactification and Chern-Simons 2-gerbes, but not actually giving an exposition.
So I went ahead and wrote such an exposition finally:
SU2-instantons from the correct maths to the traditional physics story
Beware two things:
1) this entry is meant to be included as a subsection into other entries (such as Yang-Mills instanton, BPTS instanton) therefore it is intentionally lacking toc, headlines and other introductory stuff
2) I just wrote this in one go (trying to get back to somebody waiting for me), and now I am out of steam. This hasn’t been proof-read even once yet. So unless you feel energetic about joining in the editing, better wait until a little later when this has stabilized.
Ideally this kind of account would eventually be beautified with some pictures and the like.
the entry category of sheaves had been a bit shy about giving away full information where it came to the recognition of epi/mono/isos. I have now expanded the proposition here.
Maybe somebody feels inspired to add pointer to proof, or, better yet, add proof.
contrapositive now exists. It doesn't say much (unless you were unsure which versions of this rule are intuitionistically valid, in which case, you can take a look). But there was a link to it, so I made it.
started Fierz identity to collect some references. Am still searching for the good reference for the general case…
Thanks to Manuel Bärenz for writing graded fusion category. It’s worth announcing such things here.
One strange thing with itex is that you need to separate letters for them to italicize, e.g., needs to be .
started local geometric morphism (which previousy had been just a sub-point at essential geometric morphism)
I have added at combinatorial spectrum the missing bibliographical information for Kan’s original article.
While doing so I noticed old forgotten discussion sitting there, which hereby I move from there to here:
— begin forwarded discussion —
A previous version of this entry triggered the following discussion:
+–{: .query} Mike: Are you sure about that last condition? I remember a condition more like “for each there is some finite such that all faces of in are the basepoint.
Urs: on the bottom of page 437 in the reference by Brown it says: “each simplex of has only finitely many faces different from ”.
I see that my original phrasing reflected this only very imprecisely. I have tried to improve that now. But it also seems that this condition which you mention is not implied by Brown(?) In particular, it seems this condition does not harmoize with the fact that may be negative.
But this looks like the condition which does appear in the definition of the -simplex spectra (next page of Brown). I have added that in the list of examples now.
Another question: what’s the established term for these things here? I made up both “combinatorial spectrum” and “simplicial spectrum” after reading Brown’s article, which just calls this “spectrum” without qualification. I am tending to think that “simplicial spectrum” would be a good term.
Related to that: what’s a more recent good reference on these combinatorial version of spectra?
Mike: I was remembering a condition like that from Kan’s original article “Semisimplicial spectra,” which I unfortunately don’t have access to a copy of right now. I think the idea is that a spectrum of this sort is built out of a naive prespectrum of simplicial sets (that is, a sequence of based simplicial sets with maps ) by making the -simplices of into -simplices in the spectrum. I thought the condition on is sort of saying that each simplex comes from for some . But possibly my memory is just wrong.
Since Kan’s original term was “semisimplicial spectrum” back when “semisimplicial set” meant what we now call a “simplicial set,” it’s hard to argue with “simplicial spectrum.” As far as I know, however, no algebraic topologist has really thought seriously about these things for quite some time, probably due largely to the appearance of symmetric monoidal categories of spectra (EKMM -modules, orthogonal spectra, symmetric spectra, etc.) of which there is no known analogue for this sort of spectra. It’s kind of a shame, I think, since these spectra give a really good intuition of “an object with -cells for all .” I spent a little while once trying to come up with a version of these that would have a symmetric monoidal smash product, maybe starting with simplicial symmetric spectra instead of naive prespectra, but I failed.
Urs: thanks, very useful. That’s a piece of information that I was looking for.
Yes, this combinatorial spectrum is nicely suggestive of a -category. It seems surprising that there shouldn’t be a symmetric monoidal product on that. What goes wrong?
Concerning terminology: now that I thought about it I feel that “simplicial spectrum” may tend to be misleading, as it collides with the use of “simplicial xyz” as a simplicial object internal to the category of s. Surely some people out there will already be looking at functors and call them “simplicial spectra” (?)
Mike: Yes, you’re quite right that “simplicial spectrum” should probably be reserved for a simplicial object in spectra; I wasn’t thinking. What we really need is a name for the shape category that arises here, analogous to “simplex category,” “cube category,” and so on. Like “spectrix category.” Then combinatorial spectra would be “spectricial sets.” (I’m only half joking.)
The thing that goes wrong with the symmetric monoidal product is, as far as I can tell, sort of the same thing that goes wrong for naive prespectra: there are automorphisms that don’t get taken into account. But it’s possible that no one has just been clever enough.
=–
— end forwarded discussion —
Made a start at spectral group scheme.
wrote stuff at Poincare Lie algebra
expanded associator
When I made inhabitant redirect to term a few minutes ago, I noticed a bunch of orphaned related entries.
For instance inhabited type since long ago redirect to inhabited set. I haven’t changed that yet, but I added some cross links and comments to make clear that and how the three
are related. The state of these entries deserves to be improved on, but I won’t do anything further right now.
Noticed there was no page for the left adjoint to joins. Added subtraction.
As a postscript to some discussion on virtual knot theory we had here a little under a year ago, I met Victoria Lebed yesterday and it turns out that she studied categorification questions (among other things) in her thesis! I’ve only skimmed it so far, but it seems very nice, and her approach to virtual braids is very much in the spirit of John Baez’s n-Café comment. In particular, the thesis shows that the virtual braid group is isomorphic to the group of endomorphisms , where is the free symmetric monoidal category generated by a single braided object . (The thesis also talks a lot about positive braids, which are interpreted in terms of “pre-braided” objects, i.e., an object equipped with a not necessarily invertible morphism satisfying the Yang-Baxter equation. This definition even allows for the possibility of “idempotent” braidings, which she mentioned in her talk yesterday.)
(hm, wait)
I have added something to causal structure, for the moment mainly so as to record references to definitions of causal manifolds and to proposals for axioms of local nets over these.
Added the page partition logic. Will breathe more life into it.
Created locally decomposable space.
At uniform space, we have 6 axioms defining a uniform structure (or 4 to define a covering uniformity); I had added an axiom 0 [ETA: which I probably got from Douglas Bridges] which is classically trivial but useful in constructive mathematics. I now think that it's wrong to insist on this axiom; rather, those uniform spaces that satisfy it should be singled out as particularly nice. A good word for this, which may be used in similar contexts related to generalizations of metric spaces, is ‘located’; and since there are no Google hits for "located uniform space" (except for one which says ‘neatly located’ and so does not technically conflict), I am going to use that.
The result is that there are no more references to axiom (0) at uniform space; instead, we have a definition of a located uniform space.
I thought that were references to this axiom (0) on at least on other page pointing to uniform space, but the search function doesn't find any; if you do, though, then it would be nice to fix them.
Made the chemistry page slightly less stubby.
Slightly off topic but… I’ve been working on the category of recipes as side project as a way to bring the layman to category theory, the general idea also appears in Bob Coecke’s Quantum Pictoralism. In some exchanges I pointed out to Bob that recipes are just the edible full subcategory of chemistry. Btw, here’s the first ever (buggy) visual recipe made using monoidal category theory and string diagrams.
Would anyone be offended if I added in that recipes are a full subcategory of chemistry as a sidenote?
May I also start some pages on recipes… in some way? I think it really is a great way to teach people category theory. After all, everyone eats, and if you can follow a recipe, you can do monoidal category theory.
I have expanded the list of references at supersymmetry and division algebras.
I added the definition of uniformly/almost located to located subspace, as well as attempted proofs that covert and compact subspaces of uniformly regular uniform spaces are uniformly located. Unfortunately covert sets and compact spaces — as opposed to compact locales — seem to be hard to come by constructively. It would be nice to have some more concrete examples. I also don’t know what to write for the “idea” section, nor what Toby had in mind for “topologically located”.
Dennis Borisov kindly highlights this most remarkable article to me:
First of all it establishes this table here, which I gave an entry normed division algebra Riemannian geometry – table:
| normed division algebra | Riemannian -manifolds | Special Riemannian -manifolds | |
|---|---|---|---|
| real numbers | Riemannian manifold | oriented Riemannian manifold | |
| complex numbers | Kähler manifold | Calabi-Yau manifold | |
| quaternions | quaternion-Kähler manifold | hyperkähler manifold | |
| octonions | Spin(7)-manifold | G2-manifold |
But it goes beyond that to discuss the relevant connections etc. For the moment I have included the above table in some of the relevant entries.
Had need to note down Witten’s quick argument for how brane/anti-brane annihilation suggests that D-brane charge is in K-theory: at anti-D-brane.
I gave kappa-symmetry a decent Idea-section (I hope), highlighting the super-geometric interpretation due to Sorokin et. al. Also added more references.
Urs added a picture of the Fano plane there, and I gave a description and some common notation. For some reason, whenever I wrote ($\mathbb{P}$) it turns out slanted, which is annoying. I don’t know why.
I added to continuum hypothesis a description of Easton’s theorem, and created a page on König’s theorem. The latter could stand a number of redirects due to variant spellings.
Started SVect as it was requested by various links. But I see ’sVect’ being written too. Any preference?
Started completeness theorem.
Created covert space, a concept that is to closed maps the way overts are to open maps and compacts are to proper maps. Dubuc and Penon call covert discrete topologies “compact objects” but that seems possibly misleading in general, and “covert” seems a natural analogue of “overt” when we replace “open” by “closed”. But terminological objections are welcome…
I have put the definition of this at proper map.
I’ve slowly been filling in the article on Möbius inversion away from stub status, and also created new articles for order polynomial and for zeta polynomial.
Created a stubbish entry for Newton-Cartan structure.
The entry (eso, fully faithful) factorization system gives no support for its claim. It’s not hard to see, but needs some care. Is there a reference that we could point to, for a decent proof?
since it was requested by some entries, I have created a stub entry locally cartesian closed functor
I have added two references to induction.
Also, I added the missing (!) pointer to coinduction .
I have started a bare minimum at Frobenius-Perron dimension.
I started typing into length of an object when I felt that we had an entry on this already somewhere. Where?
I’ve created iterated inductive definitions to record the definition (in response to this MO question). I also put a link to this page from ordinal analysis.
I added to allegory a section on division allegories and power allegories.
New stub cohomology of dynamical systems.
started Chern-Gauss-Bonnet theorem
Stubby start to topological topos. Will be adding material by and by.
Hello, I was curious about how to describe involutions as algebras of a monad, so I worked it out and added some simple stuff to the article involution. As always, corrections and/or generalizations are welcome.
Made a start at applicative functor.
I started a page Eff. I need to link from ’effects’ and ’handlers’, but have a query on terminology. Is it ’algebraic effect’ rather than ’algebraic side effect’ that’s more commonly used?
Not exactly in my comfort zone, but let’s hope someone will expand this stub for effect handler.
Given that I started out poking around these constructions because of monad (in linguistics), it’s interesting to find people using algebraic effects and handlers there.