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-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory 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 manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar 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 string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory 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 topicinterval category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 9th 2010

polished a bit and expanded a bit at interval category (nothing deep, just so that it looks better)

- Discussion Type
- discussion topicNoncommutative geometry and stochastic calculus
- Category Latest Changes
- Started by Eric
- Comments 6
- Last comment by Todd_Trimble
- Last Active May 8th 2010

Todd is helping me understand opposite categories beginning with $FinSet^{op}$ here.

This discussion helped prompt some improvement of the page opposite category. When I look at that page now, I see the statement:

The idea of noncommutative geometry is essentially to define a category of

*spaces*as the opposite category of a category of algebras.This reminded me of a remark I made in the “Forward” to a paper I wrote back in 2002, so I’ve now itexified that “Foreward” here:

Noncommutative Geometry and Stochastic Calculus

By the way, this also suggests that the category $FinSet$ is the category of

*spaces*opposite to the category of finite Boolean algebras in the sense of space and quantity.

- Discussion Type
- discussion topicspinors in Yang-Mills theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_van_Beek
- Last Active May 7th 2010

I noticed that recently Konrad Waldorf created a very nice article

I went through it and added definition/theorem/proof-environments and lots of hyperlinks. Some of them are unsaturated. Maybe somebody feels inspired to create corresponding entries.

- Discussion Type
- discussion topicessential geometric morphism
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active May 7th 2010

started essential geometric morphism

- Discussion Type
- discussion topicalgebra
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active May 6th 2010

the topic-cluster “algebra” is stiil a little orphaned on the Lab, I find. I added a bit to

but these entries are still a bit pitiful. Maybe somebody feels challenged to turn them into good entries. Especially the latter.

Should we have pages algebra over a Lawvere theory, algebra over a PROP, etc? Some entries look like they want such links (for instance algebra itself).

- Discussion Type
- discussion topictrig & pi
- Category Latest Changes
- Started by Michael Hardy
- Comments 7
- Last comment by Urs
- Last Active May 6th 2010

- trigonometric identities and the irrationality of pi

I created this new page. I'm new here and I have no idea whether this sort of thing belongs in a place like this.

-- naively.....

- Discussion Type
- discussion topicobject
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 6th 2010

expanded object

- Discussion Type
- discussion topicnew and revised pages on logic stuff
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by Mike Shulman
- Last Active May 6th 2010

- Recently have been adding to or creating various pages like essentially algebraic theory, Lawvere theory, Horn theory, signature, theory, and there may be others.

- Discussion Type
- discussion topicEilenberg-Zilber theorem
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 5th 2010

added the cosimplicial version of the statement to Eilenberg-Zilber theorem and included a reference that gives a proof

- Discussion Type
- discussion topicClosed functors
- Category Latest Changes
- Started by TobyBartels
- Comments 18
- Last comment by Urs
- Last Active May 4th 2010

Eric wanted to know about closed functors, so we started a page. Probably somebody has written about these before, so references would be nice, if anybody knows them. (Google gives some hits that look promising, but I can’t read them now.)

- Discussion Type
- discussion topicHousekeeping at Frolicher spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 1
- Last comment by Andrew Stacey
- Last Active May 4th 2010

I’ve done a bit of housekeeping at Froelicher space. I’ve split the page into pieces, putting each major section into its own section.

(This will necessitate a little reference chasing at manifolds of mapping spaces, and I need to put in some redirects)

I’ve put in a definition of curvaceous compactness at topological notions of Frölicher spaces. It works, but I’m not sure if it’s the right one yet.

- Discussion Type
- discussion topicenriched category theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 4th 2010

I edited the entry enriched category theory a bit, in an attempt to eventually bring that into decent shape.

I think we should eventually expand the list of related entries there and make it a floating toc.

- Discussion Type
- discussion topicexponential law
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by Mike Shulman
- Last Active May 4th 2010

It seems to me that despite so lenghty discussions and entry related to the mapping space-hm adjunction, only the ideal situations are treated (convenient categories of spaces). For this reason, I have created a new entry exponential law for spaces containing the conditions usually used in the category of ALL topological spaces, as well as few remarks about the pointed spaces.

- Discussion Type
- discussion topiccircle bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 3rd 2010

created stub for circle bundle

- Discussion Type
- discussion topiccircle group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Andrew Stacey
- Last Active May 3rd 2010

proudly presenting the circle group ;-)

- Discussion Type
- discussion topicconcrete site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 3rd 2010

created concrete site

- Discussion Type
- discussion topicgeneralized kernels
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Mike Shulman
- Last Active May 3rd 2010

Created generalized kernel, with links from a lot of places.

- Discussion Type
- discussion topicsmooth (oo,1)-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 2nd 2010

split off smooth (infinity,1)-algebra from derived smooth manifold and (infinity,1)-algebraic theory

- Discussion Type
- discussion topicCartesian fibration
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active May 2nd 2010

added more theorems to Cartesian fibration and polished the intro slightly

- Discussion Type
- discussion topiccategorification and groupoidification
- Category Latest Changes
- Started by bwebster
- Comments 10
- Last comment by bwebster
- Last Active May 2nd 2010

Taking the advice that if I write something on the internet, it should be stuck on the n-Lab, I've converted my recent comments in the n-category cafe and some old blog posts into a new page on the relationship between categorification and groupoidification: categorification via groupoid schemes

- Discussion Type
- discussion topicmanifolds of mapping spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 21
- Last comment by Eric
- Last Active May 1st 2010

Split off the mapping spaces stuff from local addition into manifolds of mapping spaces. Still plenty to do and things to check (particularly on the linear stuff, and particularly figuring out what “compact” means). Haven’t actually deleted the relevant bit from local addition yet. Also, haven’t put a table of contents at manifolds of mapping spaces since I’ve learnt that the best way to get Urs to read something is to

*not*put a toc in.

- Discussion Type
- discussion topicsub-quasi-category
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Apr 29th 2010

created sub-quasi-category

- Discussion Type
- discussion topiccograph of a profunctor
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by Urs
- Last Active Apr 29th 2010

I created cograph of a profunctor and added some references to it at cograph of a functor. All the cograph pages could probably use some unifying work.

- Discussion Type
- discussion topicinner fibration
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 29th 2010

split off inner fibration from fibrations of quasi-categories and added remarks on how it models the (oo,1)-version of the notion of cograph of a profunctor

- Discussion Type
- discussion topicCasson invariant
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by DavidRoberts
- Last Active Apr 29th 2010

Casson invariant count SU(2) local systems of integral homology spheres. Thomas considered its holomorphic counterpart which is ultimately related to counting BPS states on Calabi-Yau 3-folds.

P.S. Hmm. Tags. New option. Great. Is there a list of tags ?

- Discussion Type
- discussion topicmicrolocal analysis, D-modules, crystals
- Category Latest Changes
- Started by zskoda
- Comments 17
- Last comment by Tim_van_Beek
- Last Active Apr 28th 2010

I am in a small wave of activity along one of my principal lonegr goals in nlab: the connection between the operator theory and geometry. This is of extreme importance for physics if we ever want to go beyond the TQFTs in quantization program. As Tom Leinster has in his work seen, there are heat-kernel like expansions involved all around the place even when one takes categorical approach and the first terms are of topological nature. This is exactly so in the WKB-expansions where the zeroth term is often the exact value for topological or more general integrable models. Witten's calculation of Witten's index (related to tmf) is an example where such WKB aprpoximations are evaluated and in presence of supersymmetry there are no other terms. Thus I believe that the kantization preferred in nlab is limited to work exactly in simiklar cases and that in general we will have more terms of WKB-like nature involved. We need to develop a categorified WKB method which will unify both.

On the other hand, the WKB method is not just expansion like in quantum mechanics books, it does involve cocycles right away in usual symplectic geometry. There is so-called Maslov index related to the multidimensional WKB method which has been pioneered by V. Maslov. The quantity which is slowly changing is an analogue of the eikonal in geometric optics, so the whole thing is a generalization of the geometric optics approximation. One can see some aspects of that on (free online, on the AMS web site, under books, here) book on symplectic geometry by Guillemin and Sternberg.

Harmonic side of the stationary phase approximation (which is just a variant of WKB in fact) is studied for long under the name oscillatory integrals. This is studied especially by Lars Hörmander and the Japanese school of microlocal analysis (btw, that one is the number 3000 entry in nlab!); the differential equations are often decribed via D-modules and in nonlinear case D-schemes which Gorthendieck described via crystals.

Strangely enough Kashiwara who worked much in microlocal analysis and D-modules has created a notion of crystal bases and crystals of quantum groups but these are NOT related to crystals. Thus I created crystal basis to fix the opinions in the nlab before they go astray...

I created entry hyperfunction, one of the tools of microanalysis, by Japanese school, a neat version of generalized functions, more flexible than distributions of Laurent Schwarz. They are obtained as boundary values of holomorphic functions (reminds me of BV formalism :)).

- Discussion Type
- discussion topic(oo,1)Cat
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 28th 2010

added a section about how to compute limits and colimits in (infinity,1)Cat and in Infinity-Grpd in terms of coCartesian fibrations to these entries.

- Discussion Type
- discussion topicLocally convex space + private web
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by DavidRoberts
- Last Active Apr 28th 2010

I added a bit in the functionals section of locally convex space about coordinate projections being continuous for LCTVSs, and that there are counterexamples to this fact without local convexity. This was from memory, I hope I got it right.

I hope it’s not a fluke that I can edit from home tonight.

I also hyperlinked my front page of my web a bit, as Urs does (like it’s going out of fashion :), so I can present our model to my company, as I (and some others to whom I have explained it) would like to implement the ’open lab book’ research model we have here. I would loove to be able to do it in instiki (by which I mean the technically minded people), but we may be stuck with an awful free wiki platform, chosen for its ’minimal advertising’ (and I quote!).

Anyway, as a result, there are a bunch of new stubby pages there, that probably aren’t worth looking at yet.

- Discussion Type
- discussion topicLatest Changes: Todd
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by Todd_Trimble
- Last Active Apr 27th 2010

Sharpened up some of the discussion at finitary monad (emphasizing equivalence with Lawvere theories), and added some technical applications to reflexive coequalizer. Both were used to support a detailed proof embedded at smooth algebra.

- Discussion Type
- discussion topic(n,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 27th 2010

expanded the long-time stub entry (n,1)-topos a little more. Made Mike’s former query box an Example-section.