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    • I am constructing a table

      structure on algebras and their module categories - table

      and am including it into the relevant entries. This is a bit experimental for the moment. More details and variants should be added and maybe some of the relations stated in a better way. Help is appreciated.

    • I find the concept-formation for 2-rings in

      • Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme (arXiv:1105.3104)

      particularly clear-sighted. Among other things it improves on the rationale for considering associative algebras as 2-modules/2-vector spaces and sesquialgebras as 2-rings/3-modules/3-vector spaces.

      Where Baez-Dolan defined a “2-rig” to be a compatibly monoidal cocomplete category, theses authors observe that one should require a bit more and define a 2-ring to be a compatibly monoidal presentable category. (This follows Jacob Lurie’s discussion, some of which is alluded to at Pr(infinity,1)Cat).

      I have now written out some of the basic definitions and statements at 2-ring in a new subsection Compatibly monoidal presentable categories. I also re-organized the full Definition section a bit, adding a lead-in discussion.

    • I added some material to Puiseux series, notable the proof that for KK algebraically closed of characteristic zero, they form the algebraic closure of the field of Laurent series K((x))K((x)). This is to be connected with a number of unwritten topics like Hensel’s lemma, Newton polygon, complete local ring, and others.

      Meanwhile, I noticed that the term “local field” has, besides physics meanings, two closely related distinct mathematical meanings. One for which we have a page local field is (non-discrete) “locally compact Hausdorff topological field”, but another is “field of fractions of a complete DVR”. It’s somewhat strange that two such closely related but distinct concepts have the same name – a terrible source of confusion.

    • I found we needed an entry 2Mod such as to be able to say things like “a sesquialgebra is an algebra internal to 2Mod2Mod”.

      So I started something.

    • In End of V-valued functors, a construction is given for the end of a V-enriched functor, which references an adjunction between hom-sets and tensor products. But the article assumes only that the enrichment category V is only symmetric monoidal, not a closed monoidal, so by what right do we have this adjunction? I'm assuming that this is just an oversight and the additional assumption on V should be added (this seems to be what Kelly's book does), can you confirm?

    • the term topological subspace used to redirect to the general-purpose entry subspace. I have now instead made it redirect to subspace topology and pointed to there from subspace.

      (Also, at subspace I have removed a sentence which claimed that “On the nLab we often say ’space’ to mean ’topological space’.” Because on the contrary, on the nnLab we are dealing with general abstract mathematics and not just the small field of topology, and so we are being careful and don’t assume that “space” by default means “topological space”.)

    • created volume, just for completeness

    • I created an entry on Larry Lambe. I included a link to some (on line) notes of his on Symbolic Computation which includes discussion of the perturbation lemma from homological perturbation theory.

    • I looked again after a long while at the entry manifold structure of mapping spaces, looking for the statement that for XX a compact smooth manifold and YY any smooth manifold, the canonical Frechet structure on C ∞(X,Y)C^\infty(X,Y) coincides with the canonical diffeological structure.

      So this statement wasn’t there yet, and hence I have tried to add it, now in Properties – Relation between diffeological and Frechet manifold structure.

      To make the layout flow sensibly, I have therefore moved the material that was in the entry previously into its own section, now called Construction of smooth manifold structure on mapping space.

      While re-reading the text I found I needed to browse around a good bit to see where some definition is and where some conclusion is. So I thought I’d equip the text more with formal Definition- and Proposition environments and cross-links between them. I started doing so, but maybe I got stuck.

      Andrew, when you see this here and have a minute to spare: could you maybe check? I am maybe confused about how the {P i}\{P_i\} and {Q i}\{Q_i\} are to be read and what the index set of the charts of C ∞(M,N)C^\infty(M,N) in the end is meant to be. For instance from what you write, what forbids the choice of {P i}\{P_i\} and {Q i}\{Q_i\} being the singleton consisting just of MM and NN itself, respectively?

    • felt the need to include the following table into various entries, so I created it as an Include-file action (physics) - table

    • In light of confusion about different possible meanings, I changed cartesian functor to be largely a disambiguation page. Feel free to object.

    • stub for moment, just for completeness

    • created stub for Wick's lemma, for the moment just so as to record a pointer to a reference

    • One of the formalisms in variational calculus and in particular a formulation of classical mechanics (and also a version for geometrical optics, with eikonal in the place of principal function) is Hamilton-Jacobi equation which just got an entry.

      Eventually, I would like to transform somehow the entry classical mechanics. Namely if we fill the sections which are there written but empty, it will grow beyond usability. I think apart from introduction, the entry should have passage between various formalisms. But the details on each formalism could be better on the separate page. Now the bulk of the entry is Poisson formalism which should be I think a separate entry. But it is not easy to engineer a good plan for this yet so let us continue adding material and we can transform the overall logic later. In any case, Hamilton-Jacobi formalims should be on equal footing with Hamiltonian formalism, Lagrangean formalism, Poisson formalism, Newton formalism etc. and some exotic structures like Nambu mechanics and Routhians should be mentioned and linked, in my opinion.

    • I changed the definition at logical functor, as it said that such a thing was a cartesian functor that preserved power objects. The page cartesian functor says

      A strong monoidal functor between cartesian monoidal categories is called a cartesian functor.

      which really is only about finite products, not finite limits as Johnstone uses, which I guess is where the definition of logical functor was lifted from. So logical functor now uses the condition ’preserves finite limits’.

      So I added a clarifying remark to cartesian functor that the definition there means finite-product-preserving, and that the Elephant uses a different definition.

      However, people may wish to have cartesian functor changed, and logical functor put back how it was. I’m ok with this, but I don’t like the terminology cartesian (and I’m vaguely aware this was debated to some extent on the categories mailing list, so I am happy to go with whatever people feel strongest about).

    • It is clear that infinity-Chern-Weil theory will induce lots of examples of oo-Chern-Simons theory : for every Chern-Simons element on an ∞\infty-Lie algebroid 𝔞\mathfrak{a}, there is the corresponding generalized Chern-Simons action functional on the space of 𝔞\mathfrak{a}-valued connections/forms.

      I have started now listing all the familiar QFTs that are obtained as special cases this way. This is a joint project I am doing with Chris Rogers.

      So I started that list with comments and proofs at Chern-Simons element and began creating auxiliary entries as the need was. So there are now some stubs on

      (coupling these three yields the 2-Chern-Simons theory for the canonical invariant polynomial on a strict Lie 2-algebra !)

      also did

      (that entry was due a long time ago)

    • Created a category:reference-entry for

      • Dan Freed, 4-3-2 8-7-6, talk at ASPECTS of Topology Dec 2012

      and linked to it from some relevant entries.

    • I started discussing the Chan-Paton gauge field and how it cancels the Kapustin-part of the Freed-Witten-Kapustin anomaly for the open string.

      The technical ingredients are now all there, but I need to fill in more glue text to make this readable. Will do so, but might have to interrupt now. I ran a bit out of time here…

    • It just occurred to me that there is an immediate axiomatization of the Liouville-Poincaré 1-form (the canonical differential 1-form on a cotangent bundle) in differential cohesion.

      In fact, it is the special case of a much more general notion: for AA any type in differential cohesion the total space ∑X𝒪 X(A)\underset{X}{\sum} \mathcal{O}_X(A) of the AA-valued structure sheaf over any XX carries a canonical AA-cocycle.

      For A=Ω 1A = \Omega^1 the sheaf of 1-forms and XX a manifold, this is the traditional Liouville-Poincaré 1-form on T *XT^* X.

      I made a quick note on that at differential cohesion – Liouville-Poincaré cocycle.

      Thanks to a conversation with Owen Gwilliam I now also understand how that construction gives the antibracket in the BV-BRST complex. I still need to write that out. Not today though.

    • Just a comment, I mostly have seen k-invariant, with a lower case k. Does anyone have ‘strong’ feelings about this?

    • Since I found myself repeatedly referring to it from other nnLab entries, I finally put some content into the entry extended Lagrangian.

    • I am experimenting with a table

      But I am still experimenting. I need a table with roughly the content as given there, but loads of things still need attention. The table itself omits some details even of that which it manifestly aims to display and doesn’t display at all yet what one might also list under its title.

      Please be gentle to this stub for the moment. I need this for some lecture notes elsewhere and right now am only investing a few minutes into this, need to look into other things with higher priority for the moment. But of course eventually we should prettify this.

    • I keep drawing and re-drawing that Whitehead tower again and again. That needs to stop. So I created now an entry with a table, to be included where needed: higher spin structure - table

    • New entry IMU linked from ICM. Note that ICM page has the link to the archive of articles from old ICM-s. This is very precious as these are usually readable surveys of major contributions to mathematics covering over half a century.

    • I added a query box to the Holographic Principle page, referring to the work of Andersen and Ueno which I believe has now made rigorous that geometric quantization of Chern-Simons theory = quantum groups approach ala Reshetikhin-Turaev.

    • Before I forget, I uploaded a new version of my anafunctors paper to my page David Roberts. In particular, the finer points have been made a lot tighter. I even use technical phrases such as ’enough groupoids’ and ’admits cotensors’! :) It has also been submitted for publication.

    • Here it is: Lipschitz map. I don't know why I wrote it; I just felt like it. There really is much more to say, but I think that I've said enough for now!

    • stub for Noetherian poset

      (just needed to be able to point to it, no real content there yet)

    • added in an Examples-section to stable factorization system the statement that in an adhesive category, in particular in a topos, the (epi, mono)-factorization is stable.

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