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- Discussion Type
- discussion topicgeodesic flow
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 13th 2010

added various theorems about injectivity radius estimates and relevant literature to geodesic flow.

Important take-home message for everybody: every paracompact manifold admits a metric with positive injectivity radius.

- Discussion Type
- discussion topicskew-simplicial set
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 12th 2010

New entry skew-simplicial set with redirects crossed simplicial group, crossed simplicial set, skew-simplicial set and plurals.

- Discussion Type
- discussion topicT-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 12th 2010

added a bit more to T-algebra, but still incomplete. Need to copy over propositions and proofs from Lawvere theory

- Discussion Type
- discussion topicsmooth structure
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 11th 2010

created smooth structure

(also created differentiable manifold and added hyperlinks to the various notions in the Idea-section of manifold)

- Discussion Type
- discussion topicKleisli object
- Category Latest Changes
- Started by FinnLawler
- Comments 4
- Last comment by FinnLawler
- Last Active Oct 11th 2010

New page: Kleisli object. There’s plenty more to add yet.

- Discussion Type
- discussion topiccolouring knots
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Oct 9th 2010

If you want to divert any young minds that you know (your own for example, or some offspring or cousin or sibling or whatever) you might like to look at the colorability entry. It is sort of ’for fun’ but not completely as I hope to get on to when I’ve done some other things. (@Eric. you will have something else to do on the train! Get out your colouring pencils and a piece of paper! Find the link between 3-colourability and the symmetric group S_3. (If you know don’t tell!) You only need three pencils at the moment and as those infuriating waiters in American style restaurants say : Enjoy! :p )

- Discussion Type
- discussion topicproofs for properties of regular monomorphisms
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Oct 8th 2010

I am being bombarded by questions by somebody who is desiring details on the proofs of the statements listed at regular monomorphism, e.g. that

in Grp all monos are regular;

in Top it’s precisely the embeddings

etc.

I realize that I would need to think about this. Does anyone have a nice quick proof for some of these?

- Discussion Type
- discussion topiccategory theory
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Oct 7th 2010

This semester I have been asked to join Jaap with overlooking a handful of students who run a seminar on basic category theory.

In the course of that I will be re-looking at some nLab entries on basic stuff. Today I started looking at the cornerstone entry of the whole nLab: category theory.

I was very unhappy with that entry. Until a few minutes back. Now I am feeling a little better. That entry had consisted to a large extent (and still somewhat does) of lengthy lists of statements, all not exactly to the point, interspersed with lots of discussion with people like Todd and Toby continuously disagreeing with what somebody had written.

I think it is not sufficient to try to steer that somebody (who seems to have left us anyway). We need to rewrite this entry. If we can’t get a decent entry on category theory on the nLab, then we have no business making any claims about having a useful wiki focused on category theory.

So, I started reworking the entry:

I moved the historical remark from the very beginning to a dedicated section. An entry should start with explaining something, not with recounting how other people eventually understood that something.

After editing further the Idea section a bit, I inserted two new sections, in order to get to the main point of it all, and not bury that beneath various secondary aspects:

A section: “Basic constructions” namely universal constructions. That’s what category

*theory*is all about, after all. There is not much to be said about the concept of category itself, that’s pretty trivial. The magic is in the fact that categories support universal constructions.A section “Basic theorems”: a list of the half-dozen or so cornerstone theorems that rule category theory and mathematics as a whole. I want that nobody who glances at the entry can get away with the impression that its “just language”.

I haven’t edited much more beyond that, except that I did remove large chunks of old discussion that looked to me like mostly resolved, mostly about content that I didn’t find too exciting anyway. Should I have accidentally removed something of value, those who remember it will be able to find it in the entry’s history.

I am still not happy with the entry, but at least now I am feeling a bit better about its first third or so. I would wish a genuine category theory guru – you know who you are – would take an hour and set himself the task: here I have the chance to expose the beautiul power of category theory to the world.

- Discussion Type
- discussion topic(oo,1)-functor
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Mike Shulman
- Last Active Oct 7th 2010

in reply to a question that I received, I expanded the entry (infinity,1)-functor in various directions.

- Discussion Type
- discussion topic[circle n-bundle with connection]
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 11
- Last comment by Urs
- Last Active Oct 6th 2010

I’m confused by the definition of $\mathbf{B}^n U(1)_{diff,simp}$ at circle n-bundle with connection. Is there a “modulo $\mathbf{B}^n\mathbb{Z}$” missing? and, if so, which sense we quotient by $\mathbf{B}^n\mathbb{Z}$ there?

- Discussion Type
- discussion topicStarted "Link"
- Category Latest Changes
- Started by Andrew Stacey
- Comments 20
- Last comment by Tim_Porter
- Last Active Oct 5th 2010

Started a page at link. More to add, especially some nice pictures!, but have to go to parents’ evening now.

I’m reading Milnor’s paper “Link Groups” so shall add stuff as I read it. This should also serve as warning to a certain Prof Porter (assuming it’s the same one!) that his 1980 paper is on my list of “things to read really soon”.

- Discussion Type
- discussion topicStreet fibration
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by TobyBartels
- Last Active Oct 5th 2010

Given all the discussion on the categories list, I decided it would be worth creating Street fibration. While writing it I had occasion to put up a stub at strict 2-equivalence of 2-categories.

- Discussion Type
- discussion topicAn elementary theory of the category of topological spaces
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by DavidRoberts
- Last Active Oct 5th 2010

I tentatively added the reference

- Schlomiuk,
*An elementary theory of the category of topological spaces*

to Top.

I have to admit, though, that I did not study it. Does anyone know more about this?

- Schlomiuk,

- Discussion Type
- discussion topicComment at 'evil'
- Category Latest Changes
- Started by Andrew Stacey
- Comments 31
- Last comment by Eric
- Last Active Oct 4th 2010

André Joyal left a comment at evil, presumably sparked by the debate raging on the categories mailing list.

(Don’t remember the exact message that sparked the “debate”, but the archives for the mailing list are here).

I will admit that I’m not too enamoured of the word “evil”, but I don’t find it particularly offensive and indeed it’s “shock” value is something that I would try to retain: if you do something that is “evil” you should be darned sure that you know that you’re doing it and convinced that the final outcome justifies the means. I’m also not convinced by Joyal’s arguments about “choosing a triangulation” or whatever. Sure, we choose a triangulation to compute homology groups, but the homology groups wouldn’t be worth a dime if they actually depended on the choice of triangulation.

I also think that the “subculture” argument is vacuous.

*Every*group that has something in common could be called a “subculture” and every subculture is going to invent shortenings for referring to common terms. And of course there is great confusion when two subcultures choose the same word. My favourite story on this is when I was sitting in a garage whilst my car was being fixed. The mechanic yelled out, “You’ve got a crack in your manifold.”. I was a little confused as to what he meant! (The latest Dr Who puts a different spin on this, I believe).The thing is not to avoid being a subculture, that’s impossible, but to avoid being a clique. The distinction that I intend to draw is that cliques are defined by who they

*don’t*contain whereas subcultures are defined by who they do. Therefore anyone can join a subculture, but not anyone can join a clique.Clashes of terminology are inevitable in such a broad subject. What does the word “category” conjure to a functional analyst? Someone not well versed in algebraic geometry might ponder the meaning of a “perverse sheaf”. And the connections between limits and limits seems, if not tenuous, at least to not be all that useful in conveying intuition.

- Discussion Type
- discussion topicfundamental (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Oct 3rd 2010

created fundamental (infinity,1)-category

This is supposed to propose the evident definition. But have a critical look.

- Discussion Type
- discussion topicdisambiguation: Weil algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 3rd 2010

there are two different concepts both called “Weil algebra”. One is in Lie theory, the other is a term for duals of infinitesimally thickened points.

Promted by a question that I received, i have tried to make this state of affairs clearer on the nLab. I added a disambiguation sentence at the beginning of Weil algebra and then created infinitesimally thickened point for the other notion.

- Discussion Type
- discussion topicNew pages: isotopy and circle
- Category Latest Changes
- Started by Andrew Stacey
- Comments 18
- Last comment by Tim_Porter
- Last Active Oct 2nd 2010

Created isotopy and circle, also a bit of housekeeping (adding redirects and drop-downs) at knot and knot invariants.

For circle, my thought was to present it as an example of … just about everything! But I’m sure that there’s things I’ve missed, so the intention is that it not be a boring page “the circle is the units in $\mathbb{C}$” but rather “the circle is an example of all these different things”.

(On that thought, I’ve sometimes wondered how much of the undergraduate syllabus could be obtained by applying the centipede principle to $\mathbb{R}$.)

- Discussion Type
- discussion topicgravity - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 1st 2010

started floating TOC gravity contents and added it to the relevant pages

- Discussion Type
- discussion topicFeynman transform
- Category Latest Changes
- Started by zskoda
- Comments 9
- Last comment by TobyBartels
- Last Active Sep 30th 2010

I am still not happy with my rudimentary understanding of the characteristic classes of homotopy algebras, e.g. A-infinity algebras as presented by Hamilton and Lazarev. Kontsevich had shown how to introduce graph complexes in that setup, almost 20 years ago, but in his application to Rozansky-Witten theory he has shown the relationship to the usual Gel’fand-Fuks cohomology and usual characteristic classes of foliations. On the other hand all the similar applications are now systematized in the kind of theory Lazarev-Hamilton present. Their construction however does not seem to directly overalp but is only analogous to the usual charactersitic classes. These two points of view I can not reconcile. So I started a stub for the new entry Feynman transform. The Feynman trasnform is an operation on twisted modular operads which is Feynman graph expansion-motivated construction at the level of operads and unifies variants of graph complexes which are natural recipients of various characteristic classes of homotopy algebras.

- Discussion Type
- discussion topicrelaxed multicategory
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Sep 27th 2010

New entry relaxed multicategory. A relaxed multicategory is a special case of enriched multicategory which is accomodating for singularities like in the work of Richard Borcherds on G-vertex algebras.

- Discussion Type
- discussion topicchange of enrichment
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Sep 27th 2010

Added a mention of more general change-of-enrichment to enriched category, and a reference to Geoff Cruttwell’s thesis.

- Discussion Type
- discussion topicdoubly weak double categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by zskoda
- Last Active Sep 27th 2010

Added some discussion of other ways to define doubly-weak double categories.

- Discussion Type
- discussion topicsuitable monad
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Sep 27th 2010

New entry suitable monad. More references at enriched category.

- Discussion Type
- discussion topicPremonoidal categories
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 25th 2010

A recent question about Freyd categories on the mailing list has led me to write premonoidal category. (Freyd categories themselves are a little more obscure, and I haven’t written anything about them.)

- Discussion Type
- discussion topicFort Worth and other spurious content
- Category Latest Changes
- Started by Tim_Porter
- Comments 21
- Last comment by TobyBartels
- Last Active Sep 24th 2010

Someone has left rubbish on several pages: Fort Worth Web Design : Essays : Digital Printing : Halloween Contacts : Whitetail Deer Hunting I will go and tidy up but it is worth checking where it came from.

- Discussion Type
- discussion topiccoideal, quotient bialgebra
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Sep 23rd 2010

New entries coideal, quotient bialgebra and various redirects like left coideal, right coideal, quotient Hopf algebra. And quantum flag variety. This complements earlier today reported stubs quantum homogeneous space and Ulrich Kraehmer.

- Discussion Type
- discussion topicoo-Lie algebroid valued forms
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Sep 23rd 2010

started infinity-Lie algebroid valued differential forms , since that is needed all through our discussion of oo-Chern-Weil elsewhere. But right now the entry is stubby.

- Discussion Type
- discussion topicDefinition of pseudofunctor
- Category Latest Changes
- Started by TobyBartels
- Comments 29
- Last comment by Urs
- Last Active Sep 23rd 2010

I wrote down a definition at pseudofunctor.

- Discussion Type
- discussion topicChern-Simons circle 7-bundle
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 23rd 2010

created Chern-Simons circle 7-bundle

- Discussion Type
- discussion topicclassical Lie groups
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by zskoda
- Last Active Sep 22nd 2010

created stubs classical Lie group and exceptional Lie group and linked to them from simple Lie group (all very stubby)

- Discussion Type
- discussion topicDecategorified opposites
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 20th 2010

New article: opposite magma (including monoids, groups, rings, algebras, etc).

- Discussion Type
- discussion topicexceptional structures
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by TobyBartels
- Last Active Sep 20th 2010

without really intending to do so and certainly without having the time to do so, I ended up creating some stubs for

- Discussion Type
- discussion topicinner derivation Lie 2-algebra and Cartan calclus
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 20th 2010

following a public demand, I have

created inner derivation Lie 2-algebra,

cross-linked it with Cartan calculus and pointed out how they are related,

and started at Weil algebra a section As the CE-algebra of the L-oo algebra of inner derivations.

- Discussion Type
- discussion topicedit to [period]
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Sep 20th 2010

- Discussion Type
- discussion topicalgebraic Kan complex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 19th 2010

expanded algebraic Kan complex. Added Idea-section and Properties-section

- Discussion Type
- discussion topicPoisson sigma-model Lie algebroid Chern-Simons theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 16th 2010

When I told him about it today, Dmitry Roytenberg urged that we highlight a bit more visibly a fact that was mentioned on the nLab before, but not highlighted maybe sufficiently.

Namely by the general theory of infinity-Lie algebra cohomology we have for every $\infty$-Lie algebroid $\mathfrak{a}$ equipped with an invariant polynomial $\omega$ the corresponding Chern-Simons $CS_\omega \in W(\mathfrak{a})$ wich exhibits the transgression to the corresponding $\infty$-Lie algebroid cocycle.

If you apply this general theory to a Poisson Lie algebroid, then then Chern-Simons form that drops out is the action functional of the Poisson sigma-model.

I added this remark more visibly now to Poisson Lie algebroid, Poisson sigma-model and infinity-Lie algebroid cohomology.

- Discussion Type
- discussion topicsimplicial localization of a homotopical category
- Category Latest Changes
- Started by Harry Gindi
- Comments 4
- Last comment by Harry Gindi
- Last Active Sep 16th 2010

Following sections 34 and 35 in Dwyer-Hirschhorn-Kan-Smith (DHKS), I have begun to write up a page on hammock localization and simplicial Grothendieck construction localization. simplicial localization of a homotopical category.

- Discussion Type
- discussion topicaddition to [Georges Maltsiniotis]
- Category Latest Changes
- Started by DavidRoberts
- Comments 6
- Last comment by zskoda
- Last Active Sep 16th 2010

Mike has added to Georges Maltsiniotis a link to an English version of Maltsiniotis’ version of Grothendieck’s version of an oo-groupoid. I can’t believe I missed this on the arXiv yesterday! The French version (or rather, the French predecessor) of this paper is one of those things I wish I could sit down and read in detail and pull apart, but haven’t the time (and the language was a small, but psychological barrier).

- Discussion Type
- discussion topichomotopy algebra
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Urs
- Last Active Sep 15th 2010

Created stub homotopy algebra as this is traditional terminology unifying L, A, G, B etc infinity algebras for references and disambiguation and for operadic point of view.

- Discussion Type
- discussion topicPontrjagin duality
- Category Latest Changes
- Started by John Baez
- Comments 1
- Last comment by John Baez
- Last Active Sep 15th 2010

For some bizarre reason I added a lot of information to the page Pontrjagin duality.

- Discussion Type
- discussion topiccoherence law
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by Urs
- Last Active Sep 15th 2010

created coherence law

(was surprised to find that we didn’t have this already. Or do we?)

- Discussion Type
- discussion topicSingle-sorted definition of infinity-category
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 14th 2010

I clarified how this works in single-sorted definition of category

- Discussion Type
- discussion topic(2,1)-presheaf
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 14th 2010

created stubs for (2,1)-presheaf and hom-groupoid, just because I needed to link to them

- Discussion Type
- discussion topicBits
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 13th 2010

I separated binary digit from boolean domain.

- Discussion Type
- discussion topicCliques
- Category Latest Changes
- Started by SridharRamesh
- Comments 8
- Last comment by Urs
- Last Active Sep 12th 2010

- Growing out of my recent edit to the anafunctor article, I've created a new article on cliques.

- Discussion Type
- discussion topicgeometric definition of higher categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 11th 2010

the entry geometric definition of higher categories had been in a sorry state. I have edited it a bit in an attempt to bring it into shape

- Discussion Type
- discussion topic3-morphism
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 11th 2010

stub for 3-morphism

- Discussion Type
- discussion topicinfinity-anafunctor
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 11th 2010

created infinity-anafunctor

- Discussion Type
- discussion topicconnections on n-bundles
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 10th 2010

created (or edited)

Here the first item is always defined as a coycle in, respectively, the

And I tried to establish the same kind of link pattern for

- Discussion Type
- discussion topiccurvature
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 10th 2010

noticed that the entry curvature was in all its stubiness already a mess.

So I tried to write an Idea-section that indicates how the notion of curvature appears for embedded surfaces and then gradually generalizes to that of connections on bundles and further.

Eventually I would like to split off the section on extrinsic curvature to a separate entry extrinsic curvature and Gaussian curvature.

But not now, I need to be doind something else…

- Discussion Type
- discussion topicPoisson Lie algebroid
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 10th 2010

created Poisson Lie algebroid

- Discussion Type
- discussion topicmapping class group, ribbon graph, Jakob Nielsen
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Sep 9th 2010

New entries ribbon graph, mapping class group, Jakob Nielsen and just one reference at Mumford conjecture. Mainly to set some background for yesterday started entries graph homology, noncommutative symplectic geometry and related effort to understanding of the characteristic classes for infinity algebras in work of Andrey Lazarev and Alastair Hamilton who use variants of the Kontsevich’s construction involving graph (co)homology.

- Discussion Type
- discussion topicvertical composition
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Sep 9th 2010

created vertical composition

- Discussion Type
- discussion topicNew articles
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Andrew Stacey
- Last Active Sep 9th 2010

Several new articles have been announced on the thread Entropy, and many don’t have much to do with that subject. So I’ll just list them all here. (Many of them already have discussions on the other thread, however, so do look there.)

- Discussion Type
- discussion topicnonabelian bundle gerbe
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by DavidRoberts
- Last Active Sep 9th 2010

stub for nonabelian bundle gerbe, since I need the link at infinity-Weil theory introduction

- Discussion Type
- discussion topicPossible spam: elisabeth Constantine
- Category Latest Changes
- Started by Eric
- Comments 2
- Last comment by Andrew Stacey
- Last Active Sep 9th 2010

- Discussion Type
- discussion topicpseudo-connection
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 9th 2010

started a stub for pseudo-connection, in order to satisfy links. But now I am

*really*too tired. More tomorrow.

- Discussion Type
- discussion topicformal noncommutative symplectic geometry
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Sep 8th 2010

New entry formal noncommutative symplectic geometry within the circle of entries related to graph homology.

- Discussion Type
- discussion topicmore on string diagrams
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Sep 7th 2010

I added some more variations, links, and references to string diagram. I’m sure there are a lot more references that ought to go there.

- Discussion Type
- discussion topicLinear stuff
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Urs
- Last Active Sep 7th 2010

I expanded linear algebra and wrote linear operator.