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tried to polish 3-groupoid a little
I’m struggling to further develop the page on Schur functors, which Todd and I were building. But so far I’ve only done a tiny bit of polishing. I deleted the discussion Todd and I were having near the top of the page, replacing it by a short warning that the definition of Schur functors given here needs to be checked to see if it matches the standard one. I created a page on linear functor and a page on tensor power, so people could learn what those are. And, I wound up spending a lot of time polishing the page on exterior algebra. I would like to do the same thing for tensor algebra and symmetric algebra, but I got worn out.
In that page, I switched Alt to as the default notation for exterior algebra. I hope that’s okay. I think it would be nice to be consistent, and I think is most widely used. Some people prefer .
I have created a new entry center of an abelian category. Maybe it is superfluous as it is just a special case of a construction at center. However in this context there isa number of special theorems which I plan to enter at some point later, so maybe it is not an error to have a separate entry.
Began planar algebra for no very good reason.
I wrote the beginnings of an article real closed field. I also wrote fundamental theorem of algebra, giving the proof essentially due to Artin which applies generally to real closed fields. Lucky for me, Toby recently wrote quadratic formula! :-)
Things like this have a tendency of spawning a bunch of new articles, but I left out a bunch of potential links in these articles. Please feel free to insert some!
I wrote semialgebraic set. This should spawn other entries such as o-minimal structure.
New entry cohomological descent.
Coming back to my project of working through Stone Spaces, here are some rather blasé stubs: irreducible closed subspace, T-D-space, upper interval topology.
Some things are too simple to be simple.
Enjoy zero-divisor.
I’ve redirected the new article stuff to stuff, structure, property, because all of that stuff (pun not originally intended, but kept with delight) is already there, and it didn’t seem like the author knew about it. It doesn’t have to be that way, however, so move stuff > history back to stuff if you disagree, but then make some prominent links between the articles too.
A linear category is simply a category where every Hom(x, y) is a complex vector space and the composition of morphisms is bilinear. A *-category is a linear category that has a *-operation on each Hom(x, y) (same axioms a for a *-algebra) and a category further has a norm on each Hom(x, y) that turns it into a Banach space with and for all arrows s, t (s and t composable).
Is there already a page on the nLab that describes this structure?
the entry fibrations of quasi-categories was getting too long for my taste. I have to change my original plans about it.
Now I split off left Kan fibration from it, which currently duplicates material from this entry and from fibration fibered in groupoids. I'll see how to eventualy harmonize this a bit better.
Presently my next immediate goal is to write out as a pedagogical introduction to the notion of left/right fibration a nice detailed proof for the fact that a functor is an op-fibration fibered in groupoids precisely if its nerve is a left Kan fibration.
I wanted to do that today, but got distracted. Now I am getting too tired. So I'll maybe postpone this until tomorrow...
I added material to Young diagram, which forced me to create entries for special linear group and special unitary group. I also added a slight clarification to unitary group.
I would love it if someone who knows algebraic geometry would fix this remark at general linear group:
Given a commutative field , the general linear group (or ) is the group of invertible matrices with entries in . It can be considered as a subvariety of the affine space of square matrices of size carved out by the equations saying that the determinant of a matrix is zero.
In fact it’s ’carved out’ by the inequality saying the determinant is not zero… so its description as an algebraic variety is somewhat different than suggested above. Right???
Started on bibundles, but there seem to be a raft of competing definitions. Perhaps they're all special cases of a most general definition.
started stub for A-model
started floating table of contents
I added a bit of stuff to the Hodge theory page and started a Noncommutative Hodge theory page. I hope to expand on these pages soon.
I did a little bit of rewriting and cleaning up at reflective subcategory, in an effort to make things clearer for the neophyte. Part of the cleaning-up was to remove a query initiated by Zoran under the section Characterizations (I rewrote a bit to make the question vanish altogether).
There’s another query of Zoran at the bottom which I think was answered by Mike, but let me ask before removing it.
I can not edit Croatian black hole school, the edit button redirects me to the Homepage.
Charles Siegel created split sequence
I made some edits to C-star-algebra and representation of a C-star-algebra.
Just got the following query from Harald Hanche-Olsen about the page separation axioms. As I’ve never seen that notation before either (but agree with Harald’s comments in both parts), I’m forwarding it here so that the person who first adopted it (Toby?) or others can chip in.
I hadn’t seen the notation for a neighbourhood before, but it looks like a reasonable notation that I might want to adapt. BUT it seems more appropriate for a neighbourhood of a point rather than a neighbourhood of a set. Wouldn’t or be more appropriate for that case? What is the rationale for the usage on that page?
edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.
And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?
I think the entry-situation here deserves to be further harmonized.
Over at orthogonal subcategory problem, it’s not clear to me whether or not the “objects orthogonal to ” should be morphisms orthogonal to , or if it should mean objects of of such that is orthogonal to (where denotes the terminal object). (Hell, it could even mean objects that are the source of a map orthogonal to ). I was in the process of changing stuff to fit the first interpretation, but I rolled it back and decided to ask here.
If it should in fact be the second (or third) definition, I would definitely suggest changing the notation , which is extremely misleading, since that is the standard notation for the first notion.
added to Lie infinity-groupoid a section on Circle Lie n-groupoids, i.e. those of the form , and their relation to Cech- , Deligne-, and de Rham cohomology.
Created G-delta subset of a topological space, and its snappier redirect: G-delta. (Technically, these ought to be , I guess, which de-mathemalises to Gδ but I preferred spelling the delta out in full, does that sound okay?).
This is mainly to record a result about completely regular spaces in which every point is a G-delta subset which relates to the result I put up on sequentially compact space which in turn is related to the question of when the curvaceous topology and functional topology of a Froelicher space agree.
since it was demanded at the “counterexamples”-page, I created 3-manifold. This made me create Poincare conjecture.
I find it striking that Hamilton’s Ricci flow program and Perelman’s proof by adding the dilaton hasn’t found more resonance in the String theory community. After all, this shows a deep fact about the renormalization group flow of non-critical strings on 3-dimensional targets with gravity and dilaton background.
I once chatted with Huisken and indicated that this suggests that there is a more general interesting mathematical problem where also the Kalb-Ramond field background is taken into account. I remember him being interested, but haven’t heard that anyone in this area has extended Perelman’s method to the full massles string background content. Has anyone?
felt like creating fat simplex in parallel to Bousfield-Kan map
counterexamples in algebra inspired (and largely copied from) this MO question since MO is a daft place to put that stuff and a page on the nLab seems better. (A properly indexed database would be even better, but I don’t feel like setting such up and don’t know of the existence of such a system)
I’ve added a result to the list at sequentially compact space which is an analogue of the more well-known one about compact Hausdorff spaces. This also relates to this MO question.
added to C-star algebra a stub section on the dagger-categorical formulation
As a small step towards more information about representations of operator algebras and their physical interpretation in AQFT, I extraced states from operator algebras and added Fell’s theorem. This is a theorem that is often cited in the literature, but most times not with any specific name (often with no reference, either). But I think it is both justified and usefule to call it Fell’s theorem :-)
I got tired of making unmatched links to topological locale (aka spatial locale, or locale with enough points), so I wrote a stub.
I am trying to remove the erroneous shifts in degree by that inevitably I have been making at simplicial skeleton and maybe at truncated.
So a Kan complex is the nerve of an -groupoid iff it is -coskeletal, I hope ;-)
At truncated in the examples-section i want to be claiming that the truncation adjunction in a general (oo,1)-topos is in the case of Grpd the -adjunction on Kan complexes. But I should be saying this better.
The mass of a physical system is its intrinsic energy.
I expect that Zoran will object to some of what I have written there (if not already to my one-sentence definition above), but since I cannot predict how, I look forward to his comments.
John Baez has erased our query complaining about disgusting picture at quasigroup, and left the picture. I like the theory of quasigroups but do not like to visit and contribute to sites dominated by strange will to decorate with self-proclaimed humour which is in fact tasteless.
added to CartSp a section that lists lots of notions of (generalized) geometry modeled on this category.
expanded categorical homotopy groups in an (infinity,1)-topos
added more details to the definition of the homotopy sheaves;
added a section on how the Joyal-Jardine homotopy sheaves of simplicial presheaves are a model for that.
I keep working on the entry now titled homotopy groups in an (infinity,1)-topos (used to be "homotopy groups of an oo-stack").
The following subsections I added
Geometric Pi_oo of the terminal object in a locally connected oo-topos
Geometric Pi_0 of a genral object in a locally connected topos
Examples (meaning: general examples, neither purely categorical nor geometrical, currently a discussion of simple examples that distinguish the two notions)
This all needs more work. But I'll stop for a moment and instead start now an entry on locally n-connected (oo,1)-toposes, which I need for further discussion here.
continued from here
my proposal:
Connes fusion is used to define fusion of positive energy representations of the loop group in * Antony Wassermann, Operator algebras and conformal field theory III (arXiv) and to define elliptic cohomology in * Stephan Stolz and Peter Teichner, What is an elliptic object? (link)
and removing the query box.
New entry to support the discussion with Urs about infinitesimally thickened topos.
Some of you may remember that a while ago I had started wondering how one could characterize geometric morphisms of toposes that would exhibit as an “infinitesimal thickening” of .
Instead of coming to a defnite conclusion on this one, I worked with a concrete example that should be an example of this situation: that of the Gorthendieck toposes on the sites CartSp and ThCartSp of cartesian spaces and infinitesimally thickened cartesian spaces.
But now I went through my proofs for that situation and tried to extract which abstract properties of these sites they actually depend on. Unless I am mixed up, it seems to me now that the essential property is is a coreflective subcategory of and that in the respective adjunction
buth functors preserve covers.
So maybe it makes sense to take this as a definition: a geometric morphism of Grothendieck toposes is an infinitesimal thickening if it comes from such a coreflective embedding of sites.
Details of this, with more comments on the meaning of it all and detailed proofs, I have now typed into my page on path oo-functors in the section Infinitesimal path oo-groupoids.
added to differential bimodule the basic example, thanks to Zoran
I added a disambiguation note to conjunction, since most of the links to that page actually wanted something else. Then I changed those links to something else: logical conjunction (not yet extant).
An Internet and dictionary search suggests that there is no analogous danger for disjunction (also not yet extant).
Wrote two-sided bar construction. There is a lot to add, but I added a query box under the subsection “Delooping machines” which I’m hoping someone like Mike could answer.
Moved the subsection on Hilbert space quantum mechanics from quantum information to quantum mechanics as had been suggested by someone in a query box.
Tim van Beek has written about unbounded posets at partial order.
Where is this used?
In another thread I came up with a definition of a local isomorphism in a site, working from the definition of a local homeomorphism/diffeomorphism in Top/Diff respectively (with the open cover pretopology in both cases). Then I find that there is a page local isomorphism talking about maps in presheaf categories: such a map is a local isomorphism if becomes an isomorphism on applying the sheafification functor . To quote my definition again
Definition: Let (C,J) be a site (J a pretopology). A map is a J-local isomorphism if there are covering families and such that for each the restriction is an isomorphism onto some .
I don’t claim, in the time I have available, to understand the implications of the definition at local isomorphism. I just wonder how it relates to concrete notions like local homeomorphisms (let us work with Top and open covers as covering families). Is a local homeomorphism, after applying Yoneda, a local isomorphism? Does a local isomorphism in the image of Yoneda come from a local homeomorphism? I suspect the answer is yes. Now for the biggie: can a local isomorphism be characterised in terms as basic as my definition as quoted? With my definition one avoids dealing with functor categories (and so size issues, to some extent: is very big), so if they are equivalent, I’d like to put this somewhere.
Obviously we can take the site in my definition to be a presheaf category with the canonical pretopology or something, and potentially recover the definition at local isomorphism, but for the ease of connecting with geometric ideas, I prefer something simpler.
Any thoughts?
Created small site and large site of an object in a site, as a spin off from discussion around petit topos. The latter is so named because large site is taken for sites that happen to be large. The content of this page, however, looks as though it could go somewhere discussing sheaves.