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    • started stub for Chow group

      hoping I got this right...

    • I created the page

      h-topology

      (just a stub right now).

    • Started smooth loop space, initially just a stub. Partly to contain some bits of general theory relating to which smooth paths do I use (davidroberts) and partly to start transferring some notes on the differential topology of loop spaces over to the nLab.

      In looking for somewhere to graft it on to the current nLab tree, I encountered loop space object. It seemed to me that the smooth loop space is not a loop space object, so I commented as such (thus also creating the link to smooth loop space which was my real intent). Someone who knows these things better than I do should check this.

    • created product law, since I wanted to be able to link to it…

    • Created the page unbounded topos, and some links at topos and bounded geometric morphism.

      I’m interested in the generalisation of the construction of the unbounded topos Gl(F)Gl(F) to the general case of an inaccessible comonad GG on a bounded topos (which wlog we might as well take to be SetSet EDIT: NO, LET’S NOT). In essence, why is it unbounded? Also, what nice properties can we claim of the category of coalgebras for GG, given information about GG.

      Note also, the paper HOW LARGE ARE LEFT EXACT FUNCTORS? in TAC in 2001 seems to claim something a little stronger than Johnstone does in the Elephant, and recounted at topos, namely that the existence of lex endofunctors of set is independent of ZFC (they say something more general, but it covers this case). This is mostly a note to myself, but if others feel like looking, that would be good too.

    • You may have seen in “Recently revised” that I had edited 11-dimensional supergravity in the last days. I wanted to start a section there on the details of the action functional. But after adding some formulas, I ran out of time and just left an “under construction”-warning.

      The reason I ran out of time is that I had to first write related things with higher priority into an article we are currently preparing:

      Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory.

      Later when the dust has settled and I have more leisure, I’ll try to take care of the nnLab’s 11d sugra article again.

    • wrote something at cosmic string and by copy-and-pasting-and-changing-the-degrees added something similar to domain wall and monopole. Needs a bit more polishing, maybe.

      I am really working on writing out an abstract re-formulation of this classical theory in terms of extended TQFT with defects, but not done with that yet (and will probably be interrupted again before finishing it).

    • There used to be a warning at infinity-Lie algebra cohomology about whether or not a certain functor needs to be regarded as a derived functor in order to get the correct homotopy-theoretic interpretation of oo-Lie algebroid extensions. I think I have now spelled out at synthetic differential infinity-groupoid the required details and so I replaced that warning with a pointer to a section in that latter entry.

      All this can do with a good bit more polishing. I’ll see what I can do eventually.

    • I added a link to EoM at Lie’s three theorems, where there is a statement of the first theorem - we just had the slightly disparaging sentence

      …is today regarded as lacking a good notion of differentiable manifold.

    • stub for Tamagawa number, for the moment just so as to record the article relating it to YM theory.

    • I archive here the query box from homological algebra and finite element method.

      Eric: The appearance of homological algebra in finite element methods goes back a lot earlier than Arnold. It is covered somewhat extensively in Robert Kotiuga’s (who happened to be John Baez’s dorm room mate at MIT) PhD dissertation:

      • Kotiuga, P.R., Hodge Decompositions and Computational Electromagnetics, Ph.D. Thesis, McGill University, Montreal, Canada, 1985

      I don’t think Robert would claim to be the earliest.

      Then you can go back to 1976 with Józef Dodziuk’s classic paper:

      • Finite difference approach to the Hodge theory of harmonic forms, Amer. J. Math., 98 (1976), 79-104.

      Zoran: Surely, it is emphasised that the numerous precursors exist in the Bulletin survey of Douglas Arnold et al. quoted below. But it was not in any sense systematic till rather recently. One could expand on the history…

      I have today posted a question to MathOverflow on recent-fundamental-new-directions-in-pdes.

    • The old entry 1-groupoid was a bit vague. I have added a paragraph with a more precise description.

    • I started convex cone. While imitating the link from Jordan algebra to cone, shouldn’t there be a disambiguation between the category theoretic cone and the vector space cone? But even as it is, cone is not that clear on the relation between the cone of homotopy theory and the cone of category theory.

    • I am starting a table

      noncommutative geometry - contents

      to be included as a “floating table of contents” in the relevant entries.

      Clearly this is just a beginning. Zoran will have lots of items to add.

    • Look at what has been happening at derivator. The entry was erased and various things put there by an Anonymous Coward. Another one has reinstated the original one!! Has anyone noticed this? I was travelling about the time it happened so my usual check did not occur.

      At triangle identities something similar had started. I have rolled back.

    • Was this meant to be Planck Collaboration? I have renamed it!

    • I moved much of material from Hopf algebroid to Hopf algebroid over a commutative base, where both groupoid convolution algebras and group function algebras belong there. Most of the difference is seen already at the level of bialgebroid, the stuff about antipode in general case is to be written. Some more changes to both entries.

    • For some reason, we never had map redirect to function, but now we do. Same with mapping.

      This may or may not be the best behaviour. We might actually want a page on how people distinguish these words, such as in topology (‘map’ = continuous map but ‘function’ = function, maybe).

    • started field (physics).

      So far there is an Idea-section, a general definition with some remarks, and the beginning of a list of examples, which after the first spelled out (gravity) becomes just a list of keywords for the moment.

      More later.

    • Created binary Golay code. The construction is a little involved, and I haven’t put it in yet, because I think I can nut out a nicer description. The construction I aim to describe, in slightly different notation and terminology is in

      R. T. Curtis (1976). A new combinatorial approach to M24. Mathematical Proceedings of the Cambridge Philosophical Society, 79, pp 25-42. doi:10.1017/S0305004100052075.

    • added to some relevant entries a pointer to

    • added to Hilbert bimodule a pointer to the Buss-Zhu-Meyer article on their tensor products and induced 2-category structure.

    • I have now created relative category.

      Question: Does the transferred model structure on RelCat\mathbf{RelCat} resolve Rezk’s [2001] conjecture that the classification diagram of a model category is weakly equivalent to its simplicial localisation? The N ξN_\xi functor looks very close to computing the hammock localisation to me…

    • I have added to groupoid convolution algebra the beginning of an Examples-section titled Higher groupoid convolution algebras and n-vector spaces/n-modules.

      Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a “higher C*-algebra” version of convolution algebras of higher Lie groupoids.

      I have labelled it as “under construction” to reflect the fact that this latter aspect is a bit experimental for the moment.

      The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor

      C:Grpd→Alg b op≃2Mod C \colon Grpd \to Alg_{b}^{op} \simeq 2Mod

      (as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends double groupoids to convolution sesquialgebras, hence to 3-modules with basis (3-vector spaces).

      As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the “horizontally constant” double groupoid incarnation of BG\mathbf{B}G, while the convolution algebra of GG is the algebra of the “vertically discrete” double groupoid incarnation of BG\mathbf{B}G.

      But next, if we simply replace the bare Alg b op≃2ModAlg_b^{op} \simeq 2 Mod with the 2-category C *Alg bC^\ast Alg_b of C *C^\ast-algebras and Hilbert bimodules between them and assume (as seems to be the case) that C *C^\ast-algebraic groupoid convolution is a 2-functor

      LieGrpd ≃→C *Alg n op LieGrpd_{\simeq} \to C^\ast Alg_n^{op}

      then the same argument goes through as before and yields convolution “C *C^\ast-2-algebras” that look like Hopf-C*-algebras. Etc. Seems to go in the right direction…

    • seeing the announcement of that diffiety summer school made me think that we should have a dedicated entry titled cohomological integration which points to the aspects of this discussed already elswhere on the nLab, and which eventually lists dedicated references, if any. So I created a stub.

      Does anyone know if there is a published reference to go with the relevant diffiety-school page ?

    • dropped some lines into a new Properties-section in the old and neglected entry bibundle. But not for public consumption yet.

    • I felt we were lacking an entry closure operator. I have started one, but don’t have more time now. It’s left in a somewhat sad incomplete state for the moment.

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