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    • Tall-Wraith monoid

      Updated the reference to "The Hunting of the Hopf Ring" since it's now appeared in print.

    • I added a comment to the end of the discussion at predicative mathematics to the effect that free small-colimit completions of toposes are examples of locally cartesian closed pretoposes that are generally not toposes.

    • I added the notion of a regular curve to curve. In differential geometry, for most purposes only regular curves are useful: the parametrized smooth curves with never vanishing velocity. Smooth curves as smooth maps from the interval are not of much use without the regularity condition: their image may be far from smooth, with e.g. cusps and clustered sequences of self/intersections.

    • Do you have some ideas on how to define a general/higher notion of local Kan extension in an n-category, that gives back the usual notion in a 2-category? I am talking of local kan extension Lan_F G, with F and G two morphisms, that is given by a 2-cell with F and G on the boundary that is universal among such 2-cells.

      I would define it using as in the nlab page, the corepresentation of the functor Hom(G,F^*_) but this does not make sense in a weak n-category. I don't want of a Lurie type kan extension given by adjoint to F^*. Want something weaker.

      One could also use simply truncation to a 2-category, but is there something finer than that?

      The applications i have in mind are related to higher doctrines and theories, derived algebras and their universal properties.

      Is there in the litterature something finer than that and useful?
    • added stuff to Lie 2-group: more in the Idea-section, more examples, some constructions, plenty of references.

    • I am working on further bringing the entry

      infinity-Chern-Weil theory introduction

      into shape. Now I have spent a bit of time on the (new) subsection that exposes just the standard notion of principal bundles, but in the kind of language (Lie groupoids, anafunctors, etc) that eventually leads over to the description of smooth principal oo-bundles.

      I want to ask beta-testers to check this out, and let me know just how dreadful this still is ! ;-) The section I mean is at

      Principal n-bundles in low dimension

    • I found an interesting question on MO (here) and merrily set out to answer it. The answer got a bit long, so I thought I’d put it here instead. Since I wrote it in LaTeX with the intention of converting it to a suitable format for MO, it was simplicity itself to convert it instead to something suitable for the nLab.

      The style is perhaps not quite right for the nLab, but I can polish that. As I said, the original intention was to post it there so I started writing it with that in mind. I’ll polish it up and add in more links in due course.

      The page is at: on the manifold structure of singular loops, though I’m not sure that that’s an appropriate title! At the very least, it ought to have a subtitle: “or the lack of it”.

    • I’ve written path and loop, in order to record the misconception in the last paragraph of the definition of the latter. Along the way, I noticed that the graph-theoretic concepts are special cases of the topological ones, so enjoy.

    • I created Frobenius map, since I had linked to it in several places.

    • On well-order and elsewhere, I’ve implied that a well-order (a well-founded, extensional, transitive relation) must be connected (and thus a linear order). But this is not correct; or at least I can’t prove it, and I’ve read a few places claiming that well-orders need not be linear. So I fixed well-order, although the claim may still be on the Lab somewhere else.

      Of course, all of this is in the context of constructive mathematics; with excluded middle, the claim is actually true. I also rewrote the discussion of classical alternatives at well-order to show more popular equivalents.

    • [Jim Stasheff means to post the following message here to the nnForum, but accidentally (I think) posted it here instead (I guess because that is the forum that comes up when one googles for “nForum”)]:

      Solutions of the KZ equations are usually given in terms of assymptotic behavior in certain regions. Since the region on which the eqns are defined has a nice compactication, what is the obstruction to extending the solutions to the compactification?

    • I started a stub

      symmetric space

      and added a bit of overlapping material to quandle. I would like to talk about Lie and Jordan triple systems, but I need this introductory material first.

    • I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.

    • I made an initial foray into explaining the coalgebraic aspects of recursion schemes (following Taylor) by editing well-founded relation, by including a new section “Coalgebraic formulation”. (The title is slightly awkward when it appears just after the section “Alternative formulations”; that section was on alternative formulations which are possible in classical logic, whereas this section is on a different language for presenting the intuitionistic case. Therefore I didn’t want to make it a subsection of “Alternative formulations” as currently written.)

      Also some words there on the coalgebraic formulation of simulations.

      Edit: I decided to rename “Alternative formulations” by “Formulations in classical logic”; I hope no one minds.

    • Inspired by Tom Leinster’s recent blog posts, I have created Hausdorff metric, and added the metric-space version (sans the categorical interpretation, for now) to geodesic convexity.

    • I have started a page compactly generated model category. But this is at best “under construction” (I have added a warning). I first just wanted to record Jardine’s definition referred to there. But I find something weird at least in the notation he has (he must mean homotopy colimits in the simplicial localization instead of plain colimits on homotopy hom-sets?!), and I don’t seem to find two different autrhors that agree on all the ingredients.

      I’ll leave the entry in this unsatisfactory state for the moment. There is a warning.

    • created 2-gerbe.

      This is brief, but comes with a quick remark on the two possibilities of the definition, one as in Breen, one as in Lurie.

      (And not to speak of bundle 2-gerbes, which is really a very different definition alltogether.)

      I also edited infinity-gerbe accordingly, pointing out how Lurie’s definition is just a special case of what one would arrive at if one went with Breen and kept increasing nn.

    • I’ve moved part of my beginner’s summery on Mac Lane’s proof of the coherence theorem for monoidal categories to the nnLab.

      Surely there are many mistakes, possibly fatal: I am most worried about the naive definition I used for the syntax of arrows (sadly, I could only use my nearly zero knowledge of logic), and the part including the units and unitors. There are probably many itex errors too.

      At some point I’ve realized that it was silly to use the cumbersome arrow language as is (say, writing a string like αuvu\alpha u v u instead of α u,v,w\alpha_{u,v,w}, etc.), but to correct this required changing all figures, and this is too much for me at the moment. I also apologize for using files for figures (admiting that some of them shouldn’t even be figures), but it was too much work to do all the transition from my notes to itex in one jump.

      I hope learning by seeing this page modified by people knowing more than me (anyone here, that is :) ), but if this page seems beyond hope, I have no objections to renaming it so that it will not clutter the ’lab.

      By the way, now that I’ve re-entered the page, all figures appear with a question mark, and to the see the figure I have to press on the question-mark link (previously I’ve seen all figures appearing in the page). Is there any way to solve this problem?

    • Per the suggestion of John Baez, I took a conversation we had on Google+ and made an nLab entry out of it: universality.
    • 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 nn-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.

    • 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 opSet\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!)