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

• Spurred on by the vigourous discussion on what directed homotopy should be, I have started a page looking at some of the motivation that people have used for directed homotopy theory. The present emphasis is on Goubault’ s work about 10 years ago on modelling concurrency, ( see here Models for concurrency), and I will build new entries on various of those models, such as higher transition systems which are based on cubical sets.

I have left a query on the Models for concurrency entry since I do not really know what is meant by ’Rendezvous model’ and that description needs some work done on it.

• I expanded Levi-Civita connection:

• moved the discussion in terms of Christoffel-symbol components that had been there to its own section “In terms of Christoffel symbols”;

• stated the abstract definition clearly right at the beginning;

• stated this more in detail in “first order formalism”, i.e. in terms of a compatible ISO-connection.

• 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.

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

• 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 )

• 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?

• 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:

1. 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.

2. 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.

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

• 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?

• 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”.

• 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?

• 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.

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

• 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.

• 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 $\mathbb{R}$.)

• 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.

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

• 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.)

• 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.

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

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

• At period it was claimed that the ring of periods $P\subset \mathbb{C}$ is a subfield of $\mathbb{C}$. It is conjectured (see, e.g. wikipedia) that $1/\pi$ is not a period, and since $\pi$ is a period, $P$ is not expected to be a field. I’ve fixed this up.

• So far just a list of references.

• 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.

• 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).

• 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.

• created coherence law

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

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

• 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…