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 topicLie operad
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by zskoda
- Last Active Jul 26th 2010

stub for Lie operad

- Discussion Type
- discussion topicLinear functor - disambiguation
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by TobyBartels
- Last Active Jul 23rd 2010

We talk of a ’homogeneous linear functor’ at Goodwillie calculus, a functor which maps homotopy pushout squares to homotopy pullback squares. There are also higher degree homogeneous functors which map $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. These allow polynomial approximation in the functor calculus.

We also have linear functor and polynomial functor. I take it that these latter two are unrelated to each other, and to the functor calculus terms. I think we need some disambiguation.

Does anyone know why in the Goodwillie calculus those functors are called linear? Perhaps this helps:

At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.

A simple and familiar example of this is the Euler characteristic $e(X)$, where the local-to-global property for good decompositions takes the form $e(U \union V) = e(U) + e(V) - e(U \cap V)$. A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer–Vietoris sequence. Finally one can consider the functor $S P^{\infty}: Top \to Top$, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups $\pi_*(SP^{\infty}(X)) = H_*(X)$, the Meyer-Vietoris sequence for homology is thus a consequence of applying $\pi_*(-)$ to the homotopy pullback square.

It was the insight of Tom Goodwillie in the 1980’s that such “linear” functors $F: Top \to Top$ form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree $n$ takes appropriate sorts of $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

- Discussion Type
- discussion topichelp: what (oo,1)-colimit does this model?
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Mike Shulman
- Last Active Jul 23rd 2010

I am a bit stuck/puzzled with the following. Maybe you have an idea:

I have a group object $G$ and a morphism $G \to Q$. I have a model for the universal $G$-bundle $\mathbf{E}G$ (an object weakly equivalent to the point with a fibration $\mathbf{E}G \to \mathbf{B}G$).

I have another object $\mathbf{E}Q$ weakly equivalent to the point such that I get a commuting diagram

$\array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q }$Here $Q$ is not groupal and i write $\mathbf{E}Q$ only for the heck of it and to indicate that this is contractible and the vertical morphisms above are monic (cofibrations if due care is taken).

So I have $G$ acting on $\mathbf{E}G$ and the coequalizer of that action exists and is $\mathbf{B}G$

$G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{B}G$I can also consider the colimit $K$ of the diagram

$G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{E}Q \,.$That gives me a canonical morphism $\mathbf{B}G \to K$ fitting in total into a diagram

$\array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& K } \,.$Now here comes finally the question: I know that the coequalizer of $G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G$ is a model for the

$\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *$*homotopy colimit*over the diagramas you can imagine. But I am stuck: what intrinsic $(\infty,1)$-categorical operation is $K$ a model of?

I must be being dense….

- Discussion Type
- discussion topicTom Fiore et al new preprint
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by Urs
- Last Active Jul 23rd 2010

Fiore, Lück and Sauer have a new arXiv preprint, Euler characteristics of categories and homotopy colimits, which covers material from Tom Fiore’s talk at the Utrecht higher category theory day (and at CT2010). I added the link to that page.

- Discussion Type
- discussion topicgroupal model for universal principal infinity-bundles
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by DavidRoberts
- Last Active Jul 23rd 2010

created

groupal model for universal principal infinity-bundles

in order to record and link David Roberts’s result.

to go with this, I also created universal principal infinity-bundle.

- Discussion Type
- discussion topiccategorification via groupoid schemes
- Category Latest Changes
- Started by John Baez
- Comments 9
- Last comment by TobyBartels
- Last Active Jul 22nd 2010

- In the article categorification via groupoid schemes, I removed a distracting query box containing a discussion of how to get a double slash in TeX. The answer was that // works, but is ugly, while prettier things like \sslash may not work for people who don't have the font loaded.

- Discussion Type
- discussion topicuniversal connection on universal G-principal bundle
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by zskoda
- Last Active Jul 22nd 2010

added a stubby

to the entry Lie infinity-groupoid.

The punchline is that if we pick a groupal model for $\mathbf{E}G$ – our favorite one is the Lie 2-group $INN(G)$ – then by the general nonsense of Maurer-Cartan forms on $\infty$-Lie groups there is a Maurer-Cartan form on $\mathbf{E}G$. This is, I claim, the universal Ehresmann connection on $\mathbf{E}G$.

The key steps are indicated in the section now, but not exposed nicely. I expect this is pretty unreadable for the moment and I tried to mark it clearly as being “under construction”. But tomorrow I hope to polish it .

- Discussion Type
- discussion topicnew entry:contramodule
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Jul 21st 2010

Over a coring there are not only the left and right comodules, but also the left and right contramodules !

- Discussion Type
- discussion topiclax natural transformation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2010

I tried to polish and impove the idea-section at lax natural transformation after pointing to it from MO

- Discussion Type
- discussion topic[topological submersion]
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Jul 21st 2010

created topological submersion. I’ve seen more than one definition of this, and both could be useful. My natural inclination is to the more general, where each point in the domain has a local section through it.

On a side note I use a related condition in my thesis for a topological groupoid over a space: every object is isomorphic to one in the image of a local section. This was used in conjunction with local triviality of topological bigroupoids to define certain sorts of 2-bundles.

- Discussion Type
- discussion topicCartan calculus
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Jul 20th 2010

stub for Cartan calculus

- Discussion Type
- discussion topichypercohomology
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Kevin Lin
- Last Active Jul 20th 2010

expanding the entry hypercohomology started by Kevin Lin, I wrote an Idea-section that tries to explain the $n$POV on this

- Discussion Type
- discussion topic[Lie groupoid] and [locally trivial category]
- Category Latest Changes
- Started by DavidRoberts
- Comments 20
- Last comment by Urs
- Last Active Jul 20th 2010

Edited Lie groupoid a little, and new page: locally trivial category. There is an unsaturated link at the former, to Ehresmann’s notion of internal category, which is different to the default (Grothendieck’s, I believe). The difference only shows up when the ambient category doesn’t have all pullbacks (like Diff, which was Ehresmann’s pretty much default arena). It uses sketches, or something like them. There the object of composable arrows is given as part of the data. I suppose the details don’t make too much difference, but for Lie groupoids, it means that no assumption about source and target maps being submersions.

The latter page is under construction, and extends Ehresmann’s notion of locally trivial category/groupoid to more general concrete sites. I presume his theorem about transitive locally trivial groupoids and principal bundles goes through, it’s pretty well written.

- Discussion Type
- discussion topicsimplicial principal bundle
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 20th 2010

am splitting off simplicial principal bundle from simplicial group

- Discussion Type
- discussion topicBianchi identity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 20th 2010

created Bianchi identity.

(gave it the $\infty$-Lie theory toc, but already with the new CSS code. So as soon as that CSS code is activated on the main $n$Lab, that TOC will hide itself and become a drop-down menu. I think.)

- Discussion Type
- discussion topicFrobenius, separable, semisimple and simple algebras
- Category Latest Changes
- Started by John Baez
- Comments 10
- Last comment by John Baez
- Last Active Jul 20th 2010

I’m so sick of making mistakes about separable algebras and their relation to Frobenius algebras that I wrote a page separable algebra and added more to the page Frobenius algebra. To make these pages make sense, I needed to create pages called semisimple algebra, simple algebra, and division algebra. Also projective module.

I would love it if some experts on algebraic geometry vastly enhanced the little section about algebraic geometry in separable algebra. There’s a question there, and also a very vague sentence about etale coverings.

- Discussion Type
- discussion topichypermonoid
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by TobyBartels
- Last Active Jul 20th 2010

I created hypermonoid, polishing the comments I made in the hypermonoid thread into an article. The last subsection of the article mentions a general technique for constructing hypermonoids which ought to immediately suggest further examples to a quantum group specialist like Zoran, but I am not such a specialist. I also inserted some shameless self-promotion under References.

- Discussion Type
- discussion topiccrossed complex
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 17th 2010

finally added to crossed complex…. the definition! :-)

Also added a paragraph on what the crossed complex associated to a strict globular $\infty$-groupoid is.

- Discussion Type
- discussion topiccosmic cube
- Category Latest Changes
- Started by David_Corfield
- Comments 31
- Last comment by Mike Shulman
- Last Active Jul 17th 2010

Were we to have an entry on the cosmic cube, would people be happy with that name, or should we have something less dramatic?

- Discussion Type
- discussion topichyperring
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Todd_Trimble
- Last Active Jul 15th 2010

created hyperring

- Discussion Type
- discussion topicNonabelian Algebraic Topology
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 15th 2010

I worked on Nonabelian Algebraic Topology

made the entry “category: reference”. all about the book by Brown et al – if we feel we need a more generic entry with lower case title later, we can still split it off again

then I started adding a “Contents” section similar to what we have at Elephant and Higher Topos Theory etc., and started adding some of the content of relevance for the cosmic cube.

- Discussion Type
- discussion topicNew mathematics contents
- Category Latest Changes
- Started by TobyBartels
- Comments 27
- Last comment by zskoda
- Last Active Jul 15th 2010

I’ve added some items to mathematicscontents.

I never did much with the contents pages, so I may not have organised this in the best way.

- Discussion Type
- discussion topicTwo new pages
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jul 15th 2010

- Discussion Type
- discussion topicdiamond lemma
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jul 14th 2010

To aid the parallel discussion starting around here I created entries George Bergman, Anatolij Shirshov and diamond and in my personal lab also diamond lemma (zoranskoda). Hope Todd and others will improve.

Is it only me or the nlab is unusually slow today…

- Discussion Type
- discussion topicNew page: G-norms
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jul 13th 2010

You can turn a set into a topological abelian group by equipping it with a family of G-pseudonorms.

- Discussion Type
- discussion topic0-site
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by TobyBartels
- Last Active Jul 13th 2010

shouldn’t 0-site be named (0,1)-site?

- Discussion Type
- discussion topic(2,2)-sheaves, or, Baković's 2-espaces étalé
- Category Latest Changes
- Started by DavidRoberts
- Comments 4
- Last comment by DavidRoberts
- Last Active Jul 13th 2010

Does anyone have any notes, or know of anyone who has notes, from Igor’s Oberwolfach or Utrecht talks?

- Discussion Type
- discussion topicFreyd cover
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Todd_Trimble
- Last Active Jul 12th 2010

Began Freyd cover. What’s it for?

- Discussion Type
- discussion topic2-site
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Mike Shulman
- Last Active Jul 10th 2010

created 2-site with the material from Mike’s web (as he suggested). Added pointers to original articles by Ross Street.

- Discussion Type
- discussion topicPlethysm
- Category Latest Changes
- Started by John Baez
- Comments 55
- Last comment by Todd_Trimble
- Last Active Jul 10th 2010

I started a stub on plethysm.

Does anyone know how this mathematical term originated? I hear someone suggested it to Littlewood. But who? And why? And what’s the etymology, exactly?