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- Discussion Type
- discussion topicNotation query from separation axioms
- Category Latest Changes
- Started by Andrew Stacey
- Comments 23
- Last comment by TobyBartels
- Last Active Jun 24th 2010

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 $\stackrel\circ\ni$ 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 $\stackrel\circ\supset$ or $\stackrel\circ\supseteq$ be more appropriate for that case? What is the rationale for the usage on that page?

- Discussion Type
- discussion topicorthogonality
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Harry Gindi
- Last Active Jun 24th 2010

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.

- Discussion Type
- discussion topicPfaffian line bundle
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jun 24th 2010

created stub for Pfaffian line bundle, because I needed the link to the entry and to the single reference currently given there. Will fill in more details later today.

In the course of this I also created an extremely stubby entry fermion.

- Discussion Type
- discussion topicNo-go theorem
- Category Latest Changes
- Started by Ian_Durham
- Comments 6
- Last comment by Ian_Durham
- Last Active Jun 24th 2010

Created a stub for no-go theorem. I’d like to organize it so that Bell’s theorem, the Kochen-Specker theorem, and Gleason’s theorem are referenced from the no-go theorem entry in the QM contents. Any objections?

- Discussion Type
- discussion topic[orthogonal subcategory problem]
- Category Latest Changes
- Started by Harry Gindi
- Comments 5
- Last comment by Urs
- Last Active Jun 24th 2010

Over at orthogonal subcategory problem, it’s not clear to me whether or not the “objects orthogonal to $\Sigma$” should be morphisms orthogonal to $\Sigma$, or if it should mean objects of $X$ of $C$ such that $X\to *$ is orthogonal to $\Sigma$ (where $*$ denotes the terminal object). (Hell, it could even mean objects that are the source of a map orthogonal to $\Sigma$). 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 $\Sigma^\perp$, which is extremely misleading, since that is the standard notation for the first notion.

- Discussion Type
- discussion topicspin-statistics theorem
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Ian_Durham
- Last Active Jun 23rd 2010

stub for spin-statistics theorem. Just recording a first few references so far.

- Discussion Type
- discussion topicCircle Lie n-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 23rd 2010

added to Lie infinity-groupoid a section on Circle Lie n-groupoids, i.e. those of the form $\mathbf{B}^n U(1)$, and their relation to Cech- , Deligne-, and de Rham cohomology.

- Discussion Type
- discussion topicgroup
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Jun 23rd 2010

reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.

- Discussion Type
- discussion topicG-delta subset
- Category Latest Changes
- Started by Andrew Stacey
- Comments 2
- Last comment by TobyBartels
- Last Active Jun 23rd 2010

Created G-delta subset of a topological space, and its snappier redirect: G-delta. (Technically, these ought to be $G_\delta$, 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.

- Discussion Type
- discussion topicPoincare conjecture
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 22nd 2010

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?

- Discussion Type
- discussion topicfat simplex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 22nd 2010

felt like creating fat simplex in parallel to Bousfield-Kan map

- Discussion Type
- discussion topicCounterexamples in Algebra
- Category Latest Changes
- Started by Andrew Stacey
- Comments 17
- Last comment by Urs
- Last Active Jun 22nd 2010

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)

- Discussion Type
- discussion topicAdded a result on sequentially compact spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 5
- Last comment by Andrew Stacey
- Last Active Jun 22nd 2010

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.

- Discussion Type
- discussion topicC* algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Jun 21st 2010

added to C-star algebra a stub section on the dagger-categorical formulation

- Discussion Type
- discussion topicBell's theorem
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Jun 21st 2010

Cleaned up Bell’s theorem a bit in my ongoing effort to better organize and clean up the quantum mechanics entries.

- Discussion Type
- discussion topicFell's theorem
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 4
- Last comment by Urs
- Last Active Jun 21st 2010

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 :-)

- Discussion Type
- discussion topicTopological locales
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by Mike Shulman
- Last Active Jun 19th 2010

I got tired of making unmatched links to topological locale (aka spatial locale, or locale with enough points), so I wrote a stub.

- Discussion Type
- discussion topicenriched category theory contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2010

started floating toc enriched category theory contents and added it to relevant entries

- Discussion Type
- discussion topiccoskeleton and truncation
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 18th 2010

I am trying to remove the erroneous shifts in degree by $\pm 1$ that inevitably I have been making at simplicial skeleton and maybe at truncated.

So a Kan complex is the nerve of an $n$-groupoid iff it is $(n+1)$-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 $\infty$Grpd the $(tr_{n+1} \dashv cosk_{n+1})$-adjunction on Kan complexes. But I should be saying this better.

- Discussion Type
- discussion topicMass
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Ian_Durham
- Last Active Jun 17th 2010

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.

- Discussion Type
- discussion topiccentipede pictures
- Category Latest Changes
- Started by zskoda
- Comments 65
- Last comment by zskoda
- Last Active Jun 17th 2010

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.

- Discussion Type
- discussion topicCartSpace
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by zskoda
- Last Active Jun 17th 2010

added to CartSp a section that lists lots of notions of (generalized) geometry modeled on this category.

- Discussion Type
- discussion topiccategorical homotopy groups in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 17th 2010

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.

- Discussion Type
- discussion topichomotopy groups in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 17th 2010

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.

- Discussion Type
- discussion topic[[Connes fusion]]
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 5
- Last comment by domenico_fiorenza
- Last Active Jun 16th 2010

continued from here

my proposal:

Connes fusion is used to define fusion of positive energy representations of the loop group $\mathcal{L}SU(N)$ 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.

- Discussion Type
- discussion topicsubscheme of an Abelian category
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Jun 15th 2010

New entry to support the discussion with Urs about infinitesimally thickened topos.

- Discussion Type
- discussion topicconormal bundle
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jun 15th 2010

- Discussion Type
- discussion topicinfinitesimally thickened topos
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by zskoda
- Last Active Jun 15th 2010

Some of you may remember that a while ago I had started wondering how one could characterize geometric morphisms of toposes $E \to F$ that would exhibit $E$ as an “infinitesimal thickening” of $F$.

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 $CartSp$ is a

$CartSp \stackrel{\leftarrow }{\hookrightarrow} ThCartSp$*coreflective subcategory*of $ThCartSp$ and that in the respective adjunctionbuth 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.

- Discussion Type
- discussion topicdifferential bimodule
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 15th 2010

added to differential bimodule the basic example, thanks to Zoran

- Discussion Type
- discussion topicConjunctions
- Category Latest Changes
- Started by TobyBartels
- Comments 21
- Last comment by Todd_Trimble
- Last Active Jun 15th 2010

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).