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    • I added some discussion at terminal coalgebra that the category of trees (equivalently, the category of forests Set ω opSet^{\omega^{op}}) is a terminal coalgebra for the small-coproduct cocompletion (as endofunctor on CatCat); 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 ω op→Set\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 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.

    • New entry frame bundle. Correction at affine connection: affine connection is the principal GL n(k)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))≅U(1)H^n(M,U(1))\cong U(1) for MM a closed oriented nn-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 CC, that an object XX is connected iff hom(X,−):C→Set\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.

    • Todd has added to Grothendieck topos the statement and proof that any such is total and cototal (and I have added to adjoint functor theorem the statement that this implies that all (co)limit preserving functors between sheaf toposes have (right)left adjoints).

      I notice that we should really merge Grothendieck topos with category of sheaves. But I don’t have the energy to do this now.

    • I edited adjoint functor theorem a bit: gave it an Idea-section and a References-section and, believe it or not, a toc.

      Then I opened an Examples-section and filled in what I think is an instructive simple example: the right adjoint for a colimit preserving functor on a category of presheaves.

    • added statement and pointer to the proof of the gravitational stability of Minkowski spacetime

    • Happened to notice a question at bicartesian closed category.

      Question: don’t you need distributive bicartesian closed categories to interpret intuitionistic propositional logic? Consider the or-elimination rule

      Γ,A⊢CΓ,B⊢CΓ,A+B⊢C \frac{\Gamma, A \vdash C \qquad \Gamma,B \vdash C} {\Gamma, A + B \vdash C}

      The intepretations of the two premises will be maps of type Γ×A→C\Gamma \times A \to C and Γ×B→C\Gamma \times B \to C. Then the universal property of coproducts gets us to (Γ×A)+(Γ×B)→C(\Gamma \times A) + (\Gamma \times B) \to C, but we can’t get any farther – we need a distributivity law to get Γ×(A+B)→C\Gamma \times (A+B) \to C.

    • I completed the proof of the corollary which states that for any monad TT on SetSet, that Set TSet^T has colimits.

    • I’ve started a page on the height of a variety. This is something I’ll hopefully add a ton to later. It will probably require me to add pages on Dieudonne modules, p-divisible groups, and Witt cohomology at some point.

    • I have created an entry spinning particle

      As you can see there, so far the only point this entry is making is that the worline action functional for the ordinary Dirac spinor (such as the electrons and quarks that we all consist of) happpens to be supersymmetric. I have written a little paragraph discussing this in words a little, and then mainly collected a list of references that explain this.

      To be further expanded.

    • I have only now discovered that Gonzalo Reyes is (or has been) running a blog where he has posted lots of useful-looking notes.

      For instance in the Physics-section he has a long series of expositions on basics of differential geometry with an eye towards general relativity in terms of synthetic differential geometry. I have added pointers to this to various related entries now.

    • I added the sentence

      The factorizing morphism c→im(f)c \to im(f) is sometimes called the corestriction of ff:

      to image and made corestriction redirect to this page.

    • The entry quantum state had been a bad mess with much dubious material. Where it was not dubious, it was superceded by the parallel state in AQFT and operator algebra.

      For the time being I have mostly cleared this entry and added a pointer to state in AQFT and operator algebra. I think the best would be to delete the content of this entry entirely and merge the material from “state in AQFT and operator algebra” into here. But I am not energetic enough at this time of night to do so yet.

    • I have split off classical state as a separate entry, which was implicit in some other entries.

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