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    • New stubs N-complex (the homological algebra where d N=0d^N = 0 with N>2N\gt 2) and Michel Dubois-Violette. This interested me somewhat over a decade ago. Unfortunately, I missed the seminar talk yesterday in Zagreb by one of my colleagues, Pavle Pandžić, who found with his collaborator, very recently, that a more general and more insightful redefinition of Dirac cohomology, suggested by concrete applications in representation theory, involves the homological algebra of NN-complexes. I hope there will be some writeup soon available.

    • I made linearly independent subset to satisfy a link. I put in something about the free-forgetful adjunction and something else about constructive mathematics.

    • expanded a bit the discussion of morphisms of sites at site

    • There was a parity error at n-group; I fixed that and put in the low-dimensional examples.

    • I have created a diambiguation entry at Artin-Mazur codiagonal, as the old links at bisimplicial set lead to the entry on the codiagonal of a coproduct. I have used total simplicial set as the preferred term. Perhaps a more detailed discussion of this might be useful, but I have not got the time at the moment. (I am very slow at doing diagrams, :-( )

    • Wrote generic proof with some comments about a couple seemingly weaker versions of the axiom of choice that I've never seen mentioned anywhere before (has anyone else?). Toby and I noticed these a little bit ago while thinking about exact completions, but I just now realized that they're actually good for something: proving that the category of anafunctors between two small categories is essentially small (in the "projective" way).

    • I wrote eventuality filter, although maybe this was unnecessary, and as it was mostly already there at net. Then I took some of the logic from there and adapted it to null set.

    • created free field theory with the formalization in terms of BV-complexes by Costello-Gwilliam.

    • Does anyone know if we have a discussion, somewhere, of the theorem of Thomason linking homotopy colimits with Grothendieck constructions. I have looked in places that I thought were likely but found no trace of it, but sometimes things get buried in entries on other topics so are difficult to find.

    • New entry critics of string theory to collect the references on controversies. I think they are often rambling and vague, not technically useful s the main references we want to collect under string theory and books in string theory. I have changed the sentence in string theory about mathematical definition of parts to somewhat more precise

      But every now and then some aspect of string theory, some mathematical gadget or consequence found there is isolated and redefined independently and mathematically rigorously, retaining many features originally predicted.

      The point is that most often one does not make rigorous the way some thing is defined via string theory, but one isolates an invariant of manifolds for example and defines a similar one via completely different foundations. The typical example is quantum cohomology which is defined in geometric terms and not in terms of field theory any more.

      I have one disagreement with the entry: it says that topological quantum field theory has been discovered as part of string theory research, This is not true, TQFTs were found in 1977, 1978, 1980 articles of Albert Schwartz which had nothing to do with string theory. Only much later Atiyah’s formulation is influenced by string theory.

    • I finally gave The convenient setting of global analysis a category: reference-entry. Started adding pointers to it from the References-section of some relevant entries. But there will be many more left.

    • I moved some discussion from bicategory to weak enrichment, a new page. (Possibly it was already moved somewhere else, since Mike had already deleted it, but I couldn't find it.)

    • created

      Segal’s category

      (but did we have this already as an entry under some other name?)

      Hm, the apostrophe in the page title comes out in unicode, I didn’t create it that way…. And strange things happen now when linking to it. There is a link to this page at simplex category for instance which works just fine. But something tends to go wrong…

    • I have started to extract some of the relevant key steps from Higher Algebra into tensor product of infinity-modules.

      For me that currently mainly serves as an index for how to find those needles in the haystack. But eventually I should turn it into a more comprehensive discussion.

      (Some of this used to be over at bilinear map, but I have now moved it).

    • added to the Properties-section of reflective (infinity,1)-subcategory the statement and detailed proof of the fact that reflective (oo,1)-subcategories are precisely the full subcategories on local objects.

      This proof is really not specific to (oo,1)-categories and parallels a corresponding proof for 1-categories essentially verbatim. A similar 1-categorical proof I had once typed into geometric embedding. I should really copy either one of these versions to reflective subcategory.

    • I noticed in passing that the entry fiber had been in a sad state all along. I threw in a few more keystrokes to help it a bit. And also created cofiber.

    • I just learned about rigidification and decided to record it somewhere.

      1. I’m not sure if the title is good, because there is the notion of the rigidification of quasi-categories.

      2. Surely this notion has a higher analogue that maybe someone knows more about. Surely you could take an nn-stack and consider the nn-categorical fiber product to make a notion of inertia, and then rigidify with respect to some subgroup object inside…

    • Added to reduced scheme a characterization of reducedness by the internal language of the corresponding sheaf topos: A scheme XX is reduced iff its structure ring 𝒪 X\mathcal{O}_X is a residue field in the internal sense of Sh(X)\mathrm{Sh}(X).

    • I filled a general-abstract definition into Lie differentiation. Mainly I took the key points from the beginning of Formal moduli problems and reviewed them a notation somewhat more streamlined to Lie-theoretic reasoning. Then I added an indicaton of how differential cohesion fits in. More should be added to the entry.

      I’ll see how much time and energy I have left.

    • Just in case you see this in the Recently Revised-announcement and are wondering:

      I was beginning to extract the key steps in the construction of the (∞,2)(\infty,2)-category of A ∞A_\infty-algebras and A ∞A_\infty-bimodules internal to a suitable monoidal ∞\infty-category that is in section 4.3 of Higher Algebra.

      I have strated to make some notes in this direction at bimodule – Properties – (∞,2)-category of bimodules and at bilinear map – For ∞-modules.

      But this is taking more work than I thought and I need to postpone this until next week (and change my plans for our seminar tomorrow…). Therefore for the moment this material sits there “under construction”. Please take that into account if you look at it at all.

      (On the other hand, if anyone feels like lending a hand in completing this, I’d sure be happy about it. I’ll come back to this later this week).

    • 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.

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