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- Discussion Type
- discussion topicSati Geometric...M-branes
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by zskoda
- Last Active Aug 11th 2010

Hisham Sati in January posted a survey Geometric and topological structures related to M-branes. In a hard effort of several hours of intense work I created an entry containing hyperlinked bibliography of that article (I took LaTeX source, scraped off various LaTeX commands like bf, it, bibitem etc. and then started creating various hyperlinks). Most of the hyperlinks to the arxiv and few to the project euclid are created so far. Many items still do not have proper external links which would be very welcome. This is a very nice bibliography for something of much interest to Urs, me and some other nlabizants, and I would like to have it practical for our systematic online study.

- Discussion Type
- discussion topicPasting lemmas
- Category Latest Changes
- Started by FinnLawler
- Comments 3
- Last comment by FinnLawler
- Last Active Aug 10th 2010

Added a proof of the pasting lemma to pullback, and the corresponding lemma to comma object (also added the construction by pullbacks and cotensors there).

- Discussion Type
- discussion topicInfinitary Lawvere theories
- Category Latest Changes
- Started by TobyBartels
- Comments 15
- Last comment by TobyBartels
- Last Active Aug 7th 2010

I’ve started infinitary Lawvere theory following this n-Forum discussion.

- Discussion Type
- discussion topicNew entry on Bourn plus query on semi-Abelian categories
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by zskoda
- Last Active Aug 4th 2010

I have added a stub entry to the lab on Dominique Bourn. There are quite a few links that need developing there as the protomodular category stuff is quite rudimentary. I would need to learn more about it to fill things up so if anyone does feel they can help, please charge ahead.

- Discussion Type
- discussion topic3-Lie
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by zskoda
- Last Active Aug 3rd 2010

am starting to create stubs

and am heading for

- Discussion Type
- discussion topicQuestion about typed varietes of algebras
- Category Latest Changes
- Started by Andrew Stacey
- Comments 4
- Last comment by TobyBartels
- Last Active Aug 2nd 2010

On variety of algebras appears the sentence “(This paragraph may be original research. Probably the concept does appear in the literature but under a different name.)”. The paragraph in question is about

*typed*varieties of algebras. Looking at the history, this sentence (and indeed, the whole page!) appears to be due to Toby (Bartels).I’m curious as to what part that sentence refers to, in particular due to my interest in what I call

*graded*varieties of algebras (nomenclature coming from algebraic topology and*graded*cohomology theories), which I thought was just an example of a*heterogeneous*variety of algebras, a term that I’ve come across in the literature. Certainly the concepts feel closely related, and it took a fair amount of paper chasing to find the term “heterogeneous” (though “many-sorted” theories seemed a bit more of a common term), but despite my interest, I’m no expert and am sure I’m missing something. Problem is: I don’t know what and I don’t know how to properly formulate my question!

(Added in edit): Actually, I see that the term “multisorted” is in use on Lawvere theory.

- Discussion Type
- discussion topic(hyperconnected,localic)
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Aug 2nd 2010

- Discussion Type
- discussion topicsubnet
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by TobyBartels
- Last Active Aug 2nd 2010

The definition of “subnet” under net looked wrong to me (part of the wrongness was obvious), so I changed it so that it looks correct to me. Could someone please give independent verification?

- Discussion Type
- discussion topicProper and improper subsets
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Jul 30th 2010

I found a good constructive definition of proper subset and put it in there. Also I wrote improper subset.

Edit: also family of subsets; see below.

- Discussion Type
- discussion topicdifferential cohomology - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 29th 2010

started floating TOC differential cohomology - contents

- Discussion Type
- discussion topicdiploma higher gauge theory
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Jul 29th 2010

A semester ago I announced a possible mentorship in Zagreb if a physics student would like to take to digest and write a diploma on the basis of Baez-Schreiber work on higher gauge theory. Nobody chose the topic but the page in my personal nlab is left out from those times, and maybe it will be recycled by a future announcement, though it is questionable as I am likely to leave my present institution in few months. But in order to be functional, it is good to have also the list of literature which i just compiled, including the appended list of very advanced references so that it might serve at all levels. Suggestions and usage for your own purposes are welcome.

- Discussion Type
- discussion topicholonomy
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active Jul 29th 2010

added a

*stub*entry for holonomy.Just the bare definition, and of that even only the most naive one. Don’t have time for more. But created it anyway because I needed the link.

(Sounds a bit like like: I was young and needed the money…)

- Discussion Type
- discussion topicThe *real* real numbers have characteristic zero!
- Category Latest Changes
- Started by TobyBartels
- Comments 19
- Last comment by zskoda
- Last Active Jul 28th 2010

I wrote about generaliasations of real numbers and managed to follow one link to create characteristic.

- Discussion Type
- discussion topicautomorphism oo-Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 28th 2010

started automorphism infinity-Lie algebra with some basics from Sullivan’s old article

- Discussion Type
- discussion topicdifferential 2-crossed module
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 27th 2010

started stub for differential 2-crossed module

- 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 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 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 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 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 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?

- Discussion Type
- discussion topiclocally finitely presentable categories
- Category Latest Changes
- Started by John Baez
- Comments 17
- Last comment by Todd_Trimble
- Last Active Jul 10th 2010

- I have a query for Mike, or anyone who wants to tackle it, over at locally finitely presentable category. Mike seems to be saying that only the category of models of a
*finitary*essentially algebraic theory is locally finitely presentable, but some paper seems to suggest otherwise...

- Discussion Type
- discussion topic3-groupoid
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Eric
- Last Active Jul 10th 2010

tried to polish 3-groupoid a little

- Discussion Type
- discussion topicMore new pages
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by TobyBartels
- Last Active Jul 9th 2010

- Discussion Type
- discussion topicSchur functors
- Category Latest Changes
- Started by John Baez
- Comments 87
- Last comment by Todd_Trimble
- Last Active Jul 9th 2010

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 $\Lambda$ as the default notation for exterior algebra. I hope that’s okay. I think it would be nice to be consistent, and I think $\Lambda$ is most widely used. Some people prefer $\bigwedge$.

- Discussion Type
- discussion topicNew pages and terminology clashes
- Category Latest Changes
- Started by Stephen Britton
- Comments 22
- Last comment by Stephen Britton
- Last Active Jul 9th 2010

- Hello everyone

I am new the nForum and have been informed that my additions to the nLab have introduced terminology clashes and could disrupt the coherence of the nLab. My sincerest apologies to anyone who could be negatively effected. The new pages I introduced follow:

* AbTop

* AbTor

* Alg(T)

* Aut

* Ban

* Beh

* BiComp

* BiTop

* Bij

* BooRng

* BooSpa

* Bor

* CAT

* CAT(X)

* CPO

Also started added pages after reading the nLab page 'database of categories'.

- Discussion Type
- discussion topiccenter of an abelian category
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jul 8th 2010

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.

- Discussion Type
- discussion topicNew stub: prime numbers
- Category Latest Changes
- Started by TobyBartels
- Comments 7
- Last comment by Ian_Durham
- Last Active Jul 8th 2010

- Discussion Type
- discussion topicPlanar Algebra
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active Jul 8th 2010

Began planar algebra for no very good reason.

- Discussion Type
- discussion topicreal closed field
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Jul 6th 2010

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!

- Discussion Type
- discussion topicsemialgebraic set
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by zskoda
- Last Active Jul 6th 2010

I wrote semialgebraic set. This should spawn other entries such as o-minimal structure.

- Discussion Type
- discussion topiccohomological descent
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by zskoda
- Last Active Jul 6th 2010

New entry cohomological descent.

- Discussion Type
- discussion topicMore locale theory articles
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Jul 3rd 2010

- Discussion Type
- discussion topicNew page: nuclei
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jul 1st 2010

- Discussion Type
- discussion topicMore Stone Spaces stubs
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jul 1st 2010

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.

- Discussion Type
- discussion topicMore abstract nonsense: too simple to be simple
- Category Latest Changes
- Started by TobyBartels
- Comments 16
- Last comment by Ian_Durham
- Last Active Jul 1st 2010

Some things are too simple to be simple.

- Discussion Type
- discussion topicNew page: zero-divisors
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by TobyBartels
- Last Active Jun 30th 2010

Enjoy zero-divisor.

- Discussion Type
- discussion topicStuff
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by John Baez
- Last Active Jun 30th 2010

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.

- Discussion Type
- discussion topicDo the terms C-linear and *-categories already exist on the nLab?
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 34
- Last comment by Urs
- Last Active Jun 29th 2010

A $\mathbb{C}-$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 $\mathbb{C}-$linear category that has a *-operation on each Hom(x, y) (same axioms a for a *-algebra) and a $C^*-$category further has a norm on each Hom(x, y) that turns it into a Banach space with $s^* s = |s|^2$ and $|st| \leq |s| |t|$ for all arrows s, t (s and t composable).

Is there already a page on the nLab that describes this structure?

- Discussion Type
- discussion topicleft Kan fibration
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Jun 27th 2010

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...

- Discussion Type
- discussion topicyoung diagram, various groups
- Category Latest Changes
- Started by John Baez
- Comments 7
- Last comment by Todd_Trimble
- Last Active Jun 26th 2010

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 $k$, the

**general linear group**$GL(n,k)$ (or $GL_n(k)$) is the group of invertible $n\times n$ matrices with entries in $k$. It can be considered as a subvariety of the affine space $M_{n\times n}(k)$ of square matrices of size $n$ 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???