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

• I am starting an entry symplectic infinity-groupoid.

This is still in the making. Currently there are two things:

1. A little general indication of what this is supposed to be about;

2. A proof of an assertion that serves to justify the whole concept.

Namely, the literature already knows the concept of a symplectic groupoid. This plays a big role in Weistein’s program and in particular in geometric quantization of symplectic groupoids, which induces, among other things, a notion of geometric quantization of Poisson manifolds.

As far as I am aware (though I might not have been following the latest developments here, would be grateful for comments) it is generally expected that symplectic groupoids are formally the Lie integration of Poisson Lie algebroids, but there is no proof or even formalization of this in the literature.

In the entry I indicate such a formalization and give the respective proof.

The idea is that this is a special case of the general machine of infinity-Chern-Weil theory:

namely: the symplectic form on a symplectic Lie $n$-algebroid such as the Poisson Lie algebroid is Lie theoretically an invariant polynomial. So the $\infty$-Chern-Weil homomorphism produces a corresponding morphism from the integrating smooth $\infty$-groupoid to de Rham coefficients. This is a differential form in the world of smooth $\infty$-groupoids.

The assertion is: this comes out right. Feed a Poisson Lie algebroid with its canonical invariant polynomial into $\infty$-Chern-Weil theory, out comes the “classical” symplectic Lie groupoid.

(I do this for the case that the Poisson manifold is in fact itself symplectic, which is the only case I remember having seen discussed in earlier literature. But I think I can generalize this easily.)

• I addede a paragraph at Poincaré duality about the generalizations, and created the entry (so far only descent bibliography) Grothendieck duality; the list of examples expanded at duality. All prompted by seeing the today’s arXiv article of Drinfel’d and Boyarchenko.

• started a contents-page symplectic geometry - contents and added it as a floating toc to relevant entries (there should be more, have not included everything yet)

• started entries

collecting some evident ideas and observations, most of which we have discussed at one point or other over at the $n$Café.

• prompted by the discussion in another thread I have started creating stubs

But not much real content there so far.

• I added some discussion at terminal coalgebra that the category of trees (equivalently, the category of forests $Set^{\omega^{op}}$) is a terminal coalgebra for the small-coproduct cocompletion (as endofunctor on $Cat$); this is a special case of Adamek’s theorem. I linked to this discussion by adding a section at tree. (There is of course closely related discussion at pure set as well.)

It seems to me that the nLab is a bit thin on general matters of recursion. I’ve been looking a bit at the discussion in Paul Taylor’s book, and I am becoming partial to the general idea that in some sense coalgebras and corecursion often come first; after that one may base recursive schemata on the notion of well-founded coalgebras. For example, (ill-founded) trees are really simple conceptually, or at least have a very simple uniform description: as presheaves $\omega^{op} \to Set$. This is just a simple-minded application of Adamek’s theorem. Later, one can peek inside and gets the initial algebra for the small-coproduct completion as the category of well-founded trees, but this is by no means as simple (one can’t just apply Adamek’s theorem for constructing initial algebras – the hypotheses don’t hold here!).

• wrote something at vacuum.

I mainly wanted the link to point somewhere. I don’t claim that what I have there presently is a good discussion. So I have labeled it “under construction” for the moment.

• since it was mentioned on the category theory mailing list I went to the entry measure coalgebra and edited a bit: I have added some hyperlinks and Definition- and Proposition-environments.

Somebody who created the entry should look into this issue: currently the entry mentions a ground field right at the beginning, which however never reappears again. It’s clear that everything can be done over an arbitrary ground field, I guess, but currently this is not discussed well.

In order to satisfy links I then created

• added to dilaton the action functional of dilaton gravity (Perelman’s functional)

Also references and maybe something else, I forget.

• I shouldn’t be doing this. But in a clear case of procrastination of more urgent tasks, I created a floatic TOC string theory - contents and added it to some relevant entries.

• I notice that in recent preprints (see equation (2.1) in today’s 1108.4060) people are getting awefully close to rediscovering nonabelian 2-connections in the worldvolume theory of NS-fivebranes (but they are forgetting the associator! :-).

This follows a famous old conjecture by Witten, which says that the worldvolume theory of a bunch of fivebranes on top of each other (what physicsist call a “stack” of fivebranes) should be a nonabelian principal 2-bundle/gerbe-gauge theory. If you have followed Witten’s developments since then (with his latest on Khovanov homology) you’ll know that he is suggesting that this theory is at the very heart of a huge cluster of concepts (geometric Langlands duality and S-duality being part of it).

So I should eventually expand the entry fivebrane . I’ll start with some rudiments now, but will have to interrupt soon. Hopefully more later.

• I could not find a better title, for the new entry, unfortunately: opinions on development of mathematics (should be mainly bibiliography entry). I need some place to start collecting the titles which talk about generalities of mathematical development, what is important, what is not. This is relevant for but it is not philosophy. Not only because of traditional focus of philosophy on “bigger” things like true nature of beings, meaning, ethics, cognition and so on, but more because the latter is very opinionated in the usual sense, even politics. Though we should of course, choose those which have important content, it is useful to collect those. We can have netries like math and society, even math funding for other external things of relevance, eventually. This was quick fix as I have no time now.

• I added a little bit of material to ordered field, namely that a field is orderable iff it is a real field (i.e., $-1$ is not a sum of squares). More importantly, at real closed field, I have addressed an old query of Colin Tan:

Colin: Is it true that real closure is an adjoint construction to the forgetful functor from real closed fields to orderable fields?

by writing out a proof (under Properties) that indeed the forgetful functor from category of real closed fields and field homomorphisms to the category of real fields and field homomorphisms has a left adjoint (the real closure). Therefore I am removing this query from that page over to here.

• I have created a stub for constructible universe. I did not go through the version of the definition via definability. Now constructible sets are sets in the constructible universe. The notion of course, intentionally reminds the constructible sets in topology and algebaric geometry as exposed e.g. in the books on stratified spaces, on perverse sheaves (MacPherson e.g.) and in Lurie’s Higher Topos Theory. Now I wanted to create constructible set but I was hoping that there is a common definition for all these cases or at least logically defendable unique point of view, rather than partial similarity of definitions. I mean one always have some business of unions, complements etc. starting with some primitive family, say with open sets, or algebraic sets, or open sets relative strata etc. and inductively constructs more. Now, all the operations mentioned seem to have sense in some class of lattices. Maybe in Heyting lattices or at least in Boolean lattices. On the other hand, google spits out several references on constructible lattices *one of the authors is certain Janowitz), but the definition there is disappointing. I mean I would like that one has some sort of constructible completion of certain kind of a lattice and talk about the constructible elements as the elements of constructible completion. I am sure that the nLab community could nail the wanted common generalization down or to give a reference if the literature has it already.

• I have started an entry on proper homotopy theory. This is partially since it will be needed in discussing some parts of strong shape theory, but it may also be useful for discussing duality and various other topics, including studying non-compact spaces in physical contexts. This is especially true for non-compact manifolds. (I do not know what fibre bundles etc. look like in the proper homotopy setting!)

• Following Zoran's suggestion, i have written a short entry place to describe the different meanings of this term in arithmetic and analytic geometry.
• I added some more to Lebesgue space about the cases where $1 \lt p \lt \infty$ fails.

• I have added a page on microlocalization a la Sato and Kashiwara-Schapira. It is complementary and different of the page on microlocal analysis (the approach is more algebraic). Perhaps both should be merged. Mathematically, the theory of ind-sheaves by Kashiwara-Schapira completes the bridge between classical analysis (a la Hormander) and Sato's approach, even if these two domains have quite different aims.

I saw that there is also a page called algebraic microlocalization, but i don't like this name much (perhaps because of my ignorance): localization is allways algebraic (or sheaf theoretic), and microlocal analysis describes the corresponding thing in the analysis community.

What do you think (in particular, Zoran)?
• New entry frame bundle. Correction at affine connection: affine connection is the principal $GL_n(k)$-connection on the frame bundle of the manifold, not the connection on the tangent bundle, though the latter is a special case of the corresponding associated bundle connection. Urs, do you agree with the correction ? (I think that wikipedia, linked there, terminologically agrees).

• Added another proof (this one not using the universal coefficients theorem) of the isomorphism $H^n(M,U(1))\cong U(1)$ for $M$ a closed oriented $n$-manifold to Dijkgraaf-Witten theory.

• I added a few lines under the Idea sections of colimit and limit, trying to get across some intuition that might work for an undergraduate. I was moved to do so by reading the post of Greg Weeks at the Café and the subsequent commentary.

• I created contents of contents. I find I’m losing sight of what’s in the nLab since there’s so much of it. Urs (and others) very helpfully go around putting in these “contents” links, but even then you have to be in a section to know that it’s there, so I slurped through the database and extracted all the “contents” pages and stuck them in a single page (via includes). It’s not sorted, but my idea is to update this from the database rather than revising it by hand.

It’s just a first idea at getting some sort of overview; I imagine that this sort of thing can be done much better with some sort of graph showing how the pages link together, but this was quick and easy.

• New pages:

• locally additive space: Something I’ve been musing on for a bit: inside all these “categories of smooth object” then we have the category of manifolds sitting as a nice subcategory, but that doesn’t give a very nice intrinsic definition of a “manifold”. By that I mean that suppose you knew a category of smooth spaces and took that as your starting point, could you figure out what manifolds were without knowing the answer in advance? “locally additive spaces” are an attempt to characterise manifolds intrinsically.

• kinematic tangent space: Once out beyond the realm of finite dimensional manifolds, the various notions of tangent space start to diverge and so each acquires a name. kinematic refers to taking equivalence classes of curves. There’s a bit of an overlap here with some of the stuff on Frölicher spaces, but this applies to any (cartesian closed, cocomplete) category of smooth spaces.

Apart from a few little tweaks to do with wikilinks and entities, these were generated by my LaTeX-to-iTeX package. References and all.

• The discussion about the finitary vs infinitary case at connected object made me realize that something analogous could be said about finitary vs infinitary extensive categories themselves. I added a remark along those lines to extensive category.

• In response to a very old query at connected object, I gave a proof that in an infinitary extensive category $C$, that an object $X$ is connected iff $\hom(X, -): C \to Set$ merely preserves binary coproducts.

The proof was written in classical logic. If Toby would like to rework the proof so that it is constructively valid, I would be delighted.

• I have updated the doctrine part by a new n-categorical definition of doctrine, that i use in my book on QFT.
It is more flexible (even if more naive) than the classical notion of doctrine (2-adjunction on Cat, say), because
it contains all kinds of higher categorical operads, properads, etc... in a logical categorical spirit (theories
an semantics).

I guess some similar ideas were already known to specialist (street? etc...), but it makes things confortable
to give a nave to the general notion, in particular for pedagogical purpose and presenting things to non-specialists.
• I created Dieudonne module. What is the policy on accents. Technically this should be written Dieudonné module everywhere. There is a redirect. I also defined this in the affine case, but I’m pretty sure if you replace “affine” by “flat” everything should work still.

• I wrote Specker sequence, a topic in computability theory that also has applications to constructivism.

• started an entry twisted spin structure. So far the main point is to spell out the general abstract definition and notice that this is what Murray-Singer’s “spin gerbes” are models of.

• I added remarks on Cauchy completion to the Properties-section both at proset and poset.

Also made more explicit at poset the relation to prosets.

I notice that at proset there is a huge discussion section. It would be nice if those involved could absorb into the main text whatever stable insight there is, and move the remaining discussion to the nForum here.