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    • There is a deliberately ambiguous stub at finite-dimensional space.

      We might collect there all of the nice things about finite-dimensional spaces (for various notions of ’space’).

    • have added a paragraph tangent infinity-category – Tangent infinity topos meant to extract the argument from Joyal’s “Notes on Logoi” that the tangent ∞\infty-category of an ∞\infty-topos is an ∞\infty-topos. Then a remark on how this should imply that the tangent ∞\infty-topos of a cohesive topos is itself cohesive over the tangent base ∞\infty-topos.

      I am not making any claims tonight, just sketching an argument. Hope to come back to it tomorrow when I am awake again.

    • I’ve constructed the page p-divisible group since I need it for my height of a variety page. I have to admit that I’m incredibly embarrassed that no matter how many times I look up the words “directed” “inductive” “projective” “limit” “colimit” etc I never seem to use them correctly. All of the systems are as I showed G ν→G ν+1G_{\nu}\to G_{\nu +1} I thought this corresponded to directed, inductive, or colimit, but when I looked up inductive limit in the nlab it seemed to be indicating the opposite, so maybe some of the uses are wrong.

    • In the References-section at 2-sheaf I have added three “classical” references:

      in the 1970s Grothendieck, Giraud and then Bunge usually considered “2-sheaves” – namely category-valued stacks – by default. Also there is a good body of work on 2-sheaves realized as internal categories in the underlying 1-sheaf topos. I have added a pointer to Joyal-Tierney’s Strong stacks so far, but I think much more literature exists in this direction.

      But if one goes this internalization-route at all, what one should really do is, I think, consider weak internal categories in the (2,1)-topos over the underlying site.

      Has this been studied at all? Does anyone know how 2-categories of weak internal categories in (2,1)(2,1)-toposes relate to 2-toposes? At least under nice conditions these should be equivalent, I guess. But I want to understand this better.

    • I don't know why we never had endofunction, but we didn't; now we do.

    • I have made functional and operator primarily about the meanings of these in higher-order logic, where these terms are used exclusively and unqualified. I have accordingly split off linear functional from functional; linear operator (redirecting to linear map) was already separate from operator (which was only for disambiguation). I have also checked each incoming link to functional or operator (or a plural form) to link instead to linear functional or linear operator when appropriate.

      That said, there are such things as nonlinear functionals and operators on abstract vector spaces, things which are also not functionals or operators in the type-theoretic sense. Possibly we would want pages such as nonlinear functional and nonlinear operator to cover these. (Compare nonassociative algebra, which covers a topic more general than what is covered at associative algebra but also could not be covered at simply algebra.)

      I did not know what to do with the phrase ‘various discretised versions are interesting in finite geometries as well as numerical analysis’. Are these linear functionals, type-theoretic functionals, both, or neither?

    • With our “String Geometry Network” we have another meeting in October at the Max-Planck Institute for Mathematics in Bonn.

      In each such meeting we have, besides research talks and discussion sessions, a kind of “reading course”, something to get us all on the same page of some topic.

      This time the idea is to talk about higher supergeometry and “super-string geometry”, if you wish. I am preparing some notes to go with this, and naturally I got inclined to prepare them on the nLab. They will be developing here in the entry

      Currently there is just an introduction and then a session outline with just a few linked keywords. I’ll be developing this as days go by. Depending on which reactions I get, there might be drastic revisions, or just incremental extension. We’ll see.

    • started Yetter model, still a stub so far. Tim, I trust you will add references?! :-)

    • I added Sinh’s thesis plus a link to a scan to 2-group

    • brief entry “daseinisation”

      (Note: I am not embracing the term, I just happen to want to record that somebody proposed it.)

    • There was already a discussion of ends in the topological sense at proper homotopy. (I had never seen the term hemicompact before. I knew of σ\sigma-compact which is almost the same.)

    • I’d like to add some sentences describing a geometric understanding of the “torsion” of a connection on a Riemannian manifold to torsion. But perhaps my understanding of torsion is wrong, so I’m running it by you guys first. I wrote it down on math overflow and I’m curious what people think.

    • Added a link to an expository talk I gave on “The geometry of force” giving an elementary explanation of the classical Kaluza-Klein mechanism (i.e. the idea that geodesics on the principal bundle project down to curved trajectories on base space apparently experiencing a “force”). Following the book of Bleecker, Gauge theory and variational principles.

    • This came up in another thread, where it was not really on-topic. I want to re-post it here so that it gets due attention and maybe finds a resolution:

      I find the content at canonical morphism unsatisfactory to the extent that I am voting to replace it by something else.

      First why I find it unsatisfactory: the entry describes a proposal by Jim Dolan from some time ago for how to give a formal meaning to the colloquial use of “canonical” in mathematics, but it seems to me that

      1. nobody, not even Jim Dolan himself, ever used that in practice;

      2. it does not actually capture much at all about the colloquial use of “canonical” (see below for a proposal of mine of what a proper formalization would involve);

      3. the only almost-application mentioned at the bottom of the entry, which is about morphisms of QFTs, has nothing of the “canonical” flavor to it at all (on the contrary!), and the curious notion discussed there actually becomes natural if one instead considers QFTs with boundaries, as formalized on the last pages of Lurie’s “classification of TFT” article.

      In summary, I find that the article gives an ill-motivated definition which is actually misleading and has no support from usage/practice.

      If this sounds harsh, please take it as motivation to convince me otherwise. I’ll be happy to be convinced by a good argument and will use it to improve the entry accordingly.

      But currently I think the entry content should be replaced with something else. I would tend to think that a formalization of “canonical” should involve something as follows, instead.

      It should involve some notion of constructiveness. What is canonical, in colloquial meaning of the word, is that which we can actually construct, with given data (given terms).

      For instance for XX just any set without further information, the reason why id:X→Xid \colon X \to X is the canonical map from XX to itself is because this is the only one we can actually name, whose term we can actually construct. There are all these other maps, but we can’t actually name them with the given information.

      Or: the reason why (x,y):ℝ 2⟶ℝ(x,y) \colon \mathbb{R}^2 \longrightarrow \mathbb{R} are the canonical coordinates on the plane is because they are the two which one can actually construct given the data by which the plane ℝ 2\mathbb{R}^2 was constructed, namely the two projection maps out of the project. The reason why all the other coordinates that we might put on ℝ 2\mathbb{R}^2 are “not canonical” is that while they “exist” in the sense of existence of mathematics, we cannot actually construct them with the given data.

      I expect that somebody with more genuine type-theoretical practice can easily see what I am getting at here and maybe give it a more pronounced formulation.

    • added to global model structure on functors that theorem that the projective and the injective global model structure on functors with values in a combinatorial model category is itself again a combinatorial model category.

    • I added a section Hom-spaces between cofibrant/fibrant objects with a few lemmas and their proofs at

      (infinity,1)-categorical hom-space.

      (The proofs are intentionally very small-step and hopefully "pedagogical".)

      I also reworded the introduction part a bit and replied further in the old query box there.

      Effectively my point is: I am not overly happy with the title of that entry myself, but the alternatives proposed so far still strike me as worse. The main deficiency of the title is that it may sound a bit awkward. But it has the advantage of being fairly accuratively descriptive.

      But I won't be dogmatic about this. If there is a wide-spread desire to rename the entry, please feel free to do so.

    • created an entry Hamiltonian n-vecotr field (redirecting Hamiltonian multivector field) with the definition of the “de Donder-Weyl”-Hamiltonian flows of nn-volumes and, secondly. with some comments on how to interpret this in higher geometry. Will further expand on this second piece a little later, need to interrup now.

    • I have split off reduced phase space from covariant phase space and started to expand a bit.

      In particular I tried to highlight a bit the important point that the exact presymplectic form which is induced by any local action functional on its covariant phase space (as discussed there), still has to be equipped with equivariant structure as a U(1)-prinicipal connection in order to pass to the reduced phase space.

      This is an obvious point that however I find is glossed over in much of the literature and leads to some confusion in some places: some literature fond of the covariant phase space-construction from local action functionals will highlight that this always has exact presymplectic form and will take this as reason to disregard all the subtleties of geometric quantization, which pretty much disappear for exact (pre-)symplectic forms. The point missed in such discussion is that there is non-trivial equivarint structure on the prequantization of this presymplectic form.

      This subtlety as such is of course treated correctly in all of the mathematical literature listed at qauzntization commutes with reduction, of course. But that literature in turn doesn’t mention the important construction of covariant phase spaces from local Lagrangians.

      Therefore, if anyone can point me to references that do BOTH of the following:

      1. discuss the covariant presymplectic phase space induced form a local Lagrangian;

      2. discuss the need to put equivariant connection structure on the canonically induced globally defined presymplectic potential;

      I’d be grateful.

    • gave extended supersymmetry its own entry (the content had previously been scattered around in other entries)

    • After reading some thread on mathoverflow, i decided to create the global analytic geometry and analytic Langlands program entries. The first one is now well started by Berkovich and Poineau mainly.

      The analytic Langlands program is an idea of mine that is not very precise but i am convinced would help beginners in arithmetic geometry to understand quickly important aspects of arithmetic problems, and develop interest for global analytic geometry. The point is to understand better why Langlands program is so hard using schemes and would be easier using analytic geometry. This is also true with the use of ind-schemes in local geometric langlands and Chiral algebras that should be replaced by t-adic analytic tools.

      Creating an entry on this would help put all references on the subject and make them better available.

    • at Peierls bracket I have added a bunch of further references and have slightly expanded the Idea-section

    • I have now what should be a readable pre-version of

      This still needs a round of polishing or two. But it should be at least readable.

      If anyone is interested, have a look, be critical and try to poke holes into it.

    • Changed the page local section to discuss a slightly more general concept than the local sections of a bundle, and over a more general pretopology.

    • had need for a small table worldvolume-target supersymmetry of brane sigma-models and so I created one. Have included it into the relevant entries.

      Also created a stub for superembedding approach, the middle entry of the table.

      (An nForum issue: as of late I get to see the nForum only in its plain HTML-form, which is very inconvenient. Is this a problem just on my side, or does anyone else experience this?)

    • As we were discussing profinite completions the other day in another thread I thought I would add in some points about completed group algebras at profinite group and add some mention of pseudo compact algebras to the pre-existing entry on pseudocompact rings.

      It is not clear to me what the connection between these algebras and profinite algebras should be. These pseudocompact and related linear compact algebras use finite dimensionality instead of finiteness to get a sort of algebraic compactness condition.

    • I have Polchinski’s textbook a category:reference-entry String theory, for purposes of better linking to it

    • added little bit more to super translation Lie algebra, including a remark that it is a central extension of the superpoint, regarded as an abelian super Lie algebra.

    • I only noticed now that the discussion around equation (2.14) in

      • Ling Bao, Viktor Bengtsson, Martin Cederwall, Bengt Nilsson, Membranes for Topological M-Theory (arXiv:hep-th/0507077)

      identifies the exceptional super Lie algebra (2+2)(2+2)cocycle on 𝔰𝔦𝔰𝔬(7)\mathfrak{siso}(7) (given by the brane scan) with the “topological membrane” of “topological M-theory”.

      I added a brief remark to this extent to topological membrane and updated brane scan accodingly. Hope to be expanding on this soon…

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