Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicPoincare sphere
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active May 25th 2010

Wrote about Poincare sphere, which led to perfect group. Also added a subsection “Metrizable spaces” to metric space.

- Discussion Type
- discussion topicOnline resource
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active May 24th 2010

Added Manifold Atlas Project to Online Resources.

- Discussion Type
- discussion topictriangulable spaces
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active May 24th 2010

I added a section on triangulable spaces and PL structures to simplicial complex, but this is the type of thing which gets beyond my ken pretty quickly. My real motivation is to convince myself that a space is homeomorphic to the realization of a simplicial complex (in short, is triangulable) if and only if it is homeomorphic to the realization of a simplicial set – perhaps this seems intuitively obvious, but it should be given a careful proof, and I want such a proof to have a home in the Lab. (Tim Porter said in a related discussion that there was a relevant article by Curtis in some early issue of Adv. Math., but I am not near a university library to investigate this.)

I’ll put down some preliminary discussion here. Let $P_{fin}(X)$ denote the poset of finite

*nonempty*subsets of $X$. A simplicial complex consists of a set $V$ and a down-closed subset $\Sigma \subseteq P_{fin}(V)$ such that every singleton $\{v\}$ belongs to $\Sigma$. Thus $\Sigma$ is itself a poset, and we can take its nerve as a simplicial set. The first claim is that the realization of this nerve is homeomorphic to the realization of the simplicial complex. This I believe is or should be a basic result in the technique of subdivision. Hence realizations of simplicial sets subsume triangulable spaces.For the other (harder) direction, showing that realizations of simplicial sets are triangulable, I want a lemma: that the realization of a nerve of a poset is triangulable. Basically the idea is that we use the simplicial complex whose vertices are elements of the poset and whose simplices are subsets $\{x_1, x_2, \ldots, x_n\}$ for which we have a strictly increasing chain $x_1 \lt x_2 \lt \ldots \lt x_n$. Then, the next step would use the following construction: given a simplicial set $X$, construct the poset whose elements are

*nondegenerate*simplices (elements) of $X$, ordered $x \lt y$ if $x$ is some face of $y$. The claim would be that the realization of $X$ is homeomorphic to the realization of the nerve of this poset.All of this could very well be completely standard, but it’s hard for me to find an account of this in one place. Alternatively, my intuitions might be wrong here. Or, perhaps I’m going about it in a clumsy way.

- Discussion Type
- discussion topicVistoli, Notes on Grothendieck topologies, fibered categories and descent theory
- Category Latest Changes
- Started by Eric
- Comments 32
- Last comment by Urs
- Last Active May 24th 2010

I started adding some illustrations to my personal web related to Vistoli’s paper on descent. If you like them or have suggestions to improve them, I can maybe migrate some to nLab pages:

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory (ericforgy)

- Discussion Type
- discussion topicCurrying
- Category Latest Changes
- Started by TobyBartels
- Comments 17
- Last comment by Mike Shulman
- Last Active May 24th 2010

Todd Trimble requested currying (on the Sandbox, of all places), and I wrote it (also linking to it from closed monoidal category).

- Discussion Type
- discussion topicOperad
- Category Latest Changes
- Started by Harry Gindi
- Comments 3
- Last comment by Harry Gindi
- Last Active May 23rd 2010

So, I have some pending changes on operad that I made in the sandbox and am waiting for a go-ahead to post from the interested parties, but I was also wondering if someone would be willing to write up a follow-up to the very nice definition of an operad as a monoid in the blah blah monoidal category. That is, it seems like this should give us a very nice way to define an algebra, but I don't know how one would actually go about doing it.

- Discussion Type
- discussion topiccategory theory - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2010

rearranged a bit and expanded category theory - contents. In particular I added a list with central theorems of category theory.

- Discussion Type
- discussion topicYoneda lemma - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2010

created sidebar toc Yoneda lemma - contents.

Not yet convinced myself that I found the right subdivisions and probably forgot some entries. Please improve.

- Discussion Type
- discussion topicrepresentable presheaf
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2010

added Eric’s illustrations to the Idea-section at representable presheaf. Also added a stub-section on Definition in higher category theory.

- Discussion Type
- discussion topickinds of morphisms in 2-categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Mike Shulman
- Last Active May 21st 2010

Created faithful morphism, conservative morphism, pseudomonic morphism, and discrete morphism, and added to fully faithful morphism.

- Discussion Type
- discussion topichom-functor
- Category Latest Changes
- Started by Eric
- Comments 11
- Last comment by Eric
- Last Active May 21st 2010

On the page hom-functor, it says

There is also a

$hom(-,c) : C^{op} \to Set,$**contravariant hom-functor**where $C^{op}$ is the opposite category to $C$, which sends any object $x \in C^{op}$ to the hom-set $hom(x,c)$.

If you write it like this, should you really call it “contravariant”? When you write $C^{op}$, I thought you should call it just “functor” or “covariant”. By saying it is contravariant AND writing $C^{op}$, it seems like double counting.

I hope to add some illustrations to these pages. It is a shame there are not more illustrations on the nLab since nStuff is so amenable to nice pictures.

- Discussion Type
- discussion topicMore fun with functional analysis: complete and normable
- Category Latest Changes
- Started by Andrew Stacey
- Comments 21
- Last comment by Andrew Stacey
- Last Active May 21st 2010

Added complete topological vector space including various variants (quasi-complete, sequentially complete, and some others). Hopefully got all the redirects right!

I only have Schaefer’s book at home so couldn’t check “locally complete” - I know that Jarchow deals with this in his book. Kriegl and Michor naturally only consider it in the context of smootheology so I’m not sure what the “best” characterisation is. There’s also a notational conflict with “convenient” versus “locally complete”. As Greg Kuperberg pointed out, in some places “convenient” means “locally complete and bornological” whereas in others it means just “locally complete” (in the contexts where convenient is used the distinction is immaterial as the topology is not considered an integral part of the structure).

I added these whilst working on the expansion of the TVS relationships diagram. That brought up a question on terminology. In the diagram, we have entries “Banach space” and “Hilbert space” (and “normed space” and “inner product space”). These don’t quite work, though, as for a topological vector space the correct notion of a normed space should be

*normable*space as the actual choice of norm is immaterial for the TVS properties. I’m wondering whether or not this is something to worry about. Here’s an example of where it may be an issue: a nuclear Banach space is automatically finite dimensional. That implies that its topology can be given by a Hilbert structure. However, the Hilbertian norm may not be the one that was first thought of. But that’s a subtlety that’s tricky to convey on a simple diagram. So I’d rather have “normable” than “normed”. Does anyone else have an opinion on this?If “normable” is fine, then the important question is: what’s a better way of saying “Hilbertisable”, or “Banachable”? Length doesn’t matter here, as I’m putting the expanded names in tooltips and only using abbreviations in the diagram.

- Discussion Type
- discussion topiccodiscrete cofibration
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Mike Shulman
- Last Active May 21st 2010

I started writing something about codiscrete cofibrations, which is a nice way that many categories can be canonically equipped with proarrows. Richard Garner is visiting Chicago this week, and yesterday some of us were talking about how this construction can be made very functorial, giving a very nice way to construct functors and monads on equipments; I plan to add this to the entry as well.

- Discussion Type
- discussion topicrational homotopy theory in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active May 20th 2010

started rational homotopy theory in an (infinity,1)-topos

With just slightly more it could also be called "Lie theory in an oo,1-topos" I suppose.

if you looked at this yesterday, as it was under construction, maybe have another look: I believe I could clarify the global story a bit better.

- Discussion Type
- discussion topicQuery about finite dimensional Banach spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 12
- Last comment by Todd_Trimble
- Last Active May 20th 2010

Looking at the entry Banach spaces, I found the following in the introduction:

So every $n$-dimensional real Banach space may be described (up to linear isometry, the usual sort of isomorphism) as the Cartesian space $\mathbb{R}^n$ equipped with the $p$-norm for $1 \leq p \leq \infty$

which seems to imply that every norm on a finite dimensional Banach space is a $p$-norm for some $p$. That feels to me like a load of dingo’s kidneys. To define a norm on some $\mathbb{R}^n$ I just need a nice convex set, and there’s lots more of these than the balls of $p$-norms, surely.

Am I missing something?

- Discussion Type
- discussion topicMoonshine
- Category Latest Changes
- Started by zskoda
- Comments 25
- Last comment by zskoda
- Last Active May 20th 2010

Moonshine, intentionally with capital M as most people do follow this convention for the Monster and (Monstrous) Moonshine VOA.

- Discussion Type
- discussion topicspecial relativity
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Ian_Durham
- Last Active May 19th 2010

somebody signing as “Anonymous Coward” had created special relativity and typed in a confused paragraph (the smallest confusion being that the paragraph concerned not special but general relativity).

I removed that paragraph and quickly wrote a brief “Idea”-section . But have no time to do this justice now.

- Discussion Type
- discussion topic(oo,1)-site
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 19th 2010

created (infinity,1)-site

- Discussion Type
- discussion topicfinite limit
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active May 19th 2010

created finite limit (this was previously a redirect to finitely complete category)

- Discussion Type
- discussion topicSimple groups
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active May 19th 2010

It got announced in another category, but here it is in Latest Changes:

Todd began (and then I edited) simple group.

- Discussion Type
- discussion topicconformal group
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 19th 2010

I wrote a quick entry conformal group, just from memory. Somebody could check and expand. In fact it would not be bad to have also a separate entry on conformal and on quasiconformal mappings.

- Discussion Type
- discussion topicchiral algebra
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 19th 2010

chiral algebra and improvements to vertex operator algebra

- Discussion Type
- discussion topicduplicate page compact operators
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 8
- Last comment by zskoda
- Last Active May 18th 2010

somehow I missed that there already is a page compact operator and created compact operators. The plural is another error :-) the unsatisfied link that I used to create the page was “compact operators”. When I tried to rename it to the singular term it failed, of course. Now the page compact operators is simply superfluous, but as a non-administrator I cannot delete it…

- Discussion Type
- discussion topicsequential compactness
- Category Latest Changes
- Started by Andrew Stacey
- Comments 31
- Last comment by Andrew Stacey
- Last Active May 18th 2010

Created sequential compactness, should probably link to all these compactness variations from compact space. Not sure if I got the “iff” bit right in the relationship with compactness itself; will check it myself if no-one fixes it in the meantime.

I decided that this was the key property in manifolds of mapping spaces and to stop trying to figure out a Froelicher version of sequentially compact for the time-being.

- Discussion Type
- discussion topictwist
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active May 17th 2010

the term “twist” or “twisted” is one of those insanely ambiguous terms in math. Trying to follow our recent agreement on how to deal with ambiguous page names, I tried to indicate this at twist .

- Discussion Type
- discussion topicLax-idempotent monads
- Category Latest Changes
- Started by FinnLawler
- Comments 10
- Last comment by Mike Shulman
- Last Active May 17th 2010

Created lax-idempotent 2-monad, with some definitions from Kelly–Lack. I think Kock has a couple of others. I’ll add more, like proofs of the equivalence of the definitions, and more on the cocompletion example, later (next week, probably).

- Discussion Type
- discussion topicFinal version of thesis
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by Harry Gindi
- Last Active May 17th 2010

The final copy of my thesis is up on the lab. Available from Fundamental Bigroupoids and 2-Covering Spaces. I’ve fixed the typo in definition 5.1 that made it into the print copy ;)

Thus I’ve updated the links at David Roberts, the above linked page, and on my private web home page. If anyone knows of any other places it is linked, please let me know, or update the link to point to DMRthesis_final.pdf, instead of DMR_thesis.pdf.

Now to all the other projects that are on the back burner, time permitting…

- Discussion Type
- discussion topicIntroductions to category theory in physics
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Ian_Durham
- Last Active May 17th 2010

started a section Introductions to category theory in physics at the woefully imperfect entry higher category theory and physics. So far this contains mostly th expository articles by Bob Coecke.

- Discussion Type
- discussion topiccategory with duals
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 16th 2010

I see that Peter Selinger edited and added material to category with duals

- Discussion Type
- discussion topic2-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 16th 2010

stub for 2-topos (mostly so that the links we have to it do point somewhere at least a little bit useful)