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    • created a table of contents idempotents - contents and included it as a floating TOC into the relevant entries

    • While writing at k-morphism, I noticed there is no article on globular operad (aka Batanin operad), so I wrote one. Experts please look over, and improve if desired.

    • While writing the new Idea-section now at Segal condition I felt the need to have a table of contents

      So I started one and added it to the relevant entries as a floating TOC.

    • I think we need a floating table of contents categories of categories - contents to connect our entries on related topics. I have started one.

      But this needs to be further expanded. also haven’t included it into the relevant entries yet, no time right now.

    • I have written out in some detail the proof at Grothendieck spectral sequence.

      But I still need to go through it and proof-read and polish. Handle with care for the moment. Maybe the whole thing needs to be rearranged, for readability.

    • I (only) now realize that I pretty much missed the story of familial regularity and exactness. But also it was easy to miss, with the entries that are unified by this not pointing back to it.

      To rectify this I have created now a floating TOC and am including it into all the relevant entries:

      Please check out that TOC and edit/modify as need be.

    • I've done a tiny bit of work to add a more intuitive introduction to the concepts of model category and Quillen equivalence, and I plan to do some more. If anyone wants to help, that would be great. For example, it would be nice to give some general intuition for fibrations, cofibrations and weak equivalences and why they matter.

    • slightly restructured, added table of contents and then added remarks to cobordism hypothesis (in the section "remarks") using material from blog discussion over at SecretBloggingSeminar.

    • I did some editing over at free module, under the section on submodules of free modules. I don’t have Rotman’s book before me, so I can’t check whether he assumes the commutativity hypothesis for proposition 2, but I put it in to be safe. (Actually, I’ll bet it’s needed, since we have to be careful around invariant basis number which holds for commutative rings.) The proof that I added does use this hypothesis.

      Also, I deleted the remark that this is the Nielsen-Schreier theorem in the case R=ℤR = \mathbb{Z}, since NS refers to groups as opposed to abelian groups.

    • created an entry titled Topological Quantum Field Theories from Compact Lie Groups

      on the recent (or not so recent anymore) article by Freed-Hopkins-Lurie-Teleman (therefore the capizalization).

      I typed into this a summary of their central proposal for how to formalized the path integral quantization for "direcrete" quantum field theories, in terms of higher category theory.

      I think this is important, and is actually a simple idea, but few people having looked at the article maybe get away with the take-home message here. So I tried to amplify this.

      I also have some own thoughts about this. So I put a big query box in the end, with a question.

    • At principal ideal domain, I stated and proved the theorem that for modules over a pid, submodules of free modules are free (assuming the axiom of choice), and gave a couple of corollaries. This is at the head of a section on the structure theory of modules, which obviously could be expanded to treat the structure theory of finitely generated modules over a pid.

    • Copying old query box here from pseudofunctor (having incorporated its content into the entry):

      Tim: in specifying a pseudo functor FF you have to decide whether the isomorphism goes from F(gf)F(g f) to F(g)F(f)F(g) F(f) or in the other direction. Of course they are equivalent as each will be inverse to the other. You might say that one is lax and pseudo the other op-lax and pseudo. When specifying the Grothendieck construction for such a functor, which is to be preferred?

      Both are about equally represented in the literature that I have seen which gets confusing. (In other words, I’m confused!)

      Toby: As you suggest, the two versions are equivalent, so in a way it doesn't make a difference. But it might be nice to settle a convention in case we need it.

      Tim: I have been using (for the Menagerie) the idea that there are pseudofunctors presented in two equivalent flavours lax pseudofunctor and oplax ones.

      Mike: Well, the natural comparison maps that you get in a Grothendieck fibration go in the “lax” direction F(g)F(f)→F(gf)F(g) F(f) \to F(g f), since they are induced by the universal property of cartesian arrows. In particular, if you have a functor with “weakly cartesian” liftings that don’t compose, then it is a lax functor. Not a very strong argument, but if we just want some convention it might be a reason to pick lax. I think that making too big a deal out of the difference would be misleading, though.

    • D-geometry and Riemann-Hilbert problem. In order to make more visible one of the principal directions, where the series of entries which I am writing these days is heading to.

    • I created adequate subcategory. However, once I’d done so then I found it linked from dense functor and after reading that I wasn’t sure I ought to have created the original page. I did so because I wanted to record Isbell’s idea as it’s fairly relevant to categories of generalised smooth spaces - the test spaces form an adequate subcategory (or sort of do, I need to work out the details).

      It seems to be old terminology (reading dense functor) so maybe a page devoted to it isn’t right. I could shift it to dense functor?

    • Maybe I am looking at the wrong places: is there somewhere a discussion of examples for classes of toposes that satisfy COSHEP?

      What is known about which sites induce toposes that validate COSHEP?

    • Extended the entry Cohn localization now starting with the ring viewpoint. Urs: I hope you will now agree that it is justified to call it a localization of a ring R→Σ −1RR\to \Sigma^{-1} R.

    • On some pages it is desireable to have cardinalities “ℵ\aleph” be provided with a link to their explanation. I have cerated a redirect-page ℵ for that purpose.

    • I started creating the following tables for the entry geometry of physics. After having created them there I found that these deserve to be put into the related entries, too. So therefore I put them into their own pages now and included them in related entries via

        [[!include .... - table]]

      These are the tables that I have so far:

      These need a bit mor attention. But I have to quit now for the time being. Also, I am afraid I may be running here again against Mike’s preference for notation here and there.

      But I am not dogmatic about this, I just created these tables as they happened to occur to me. I try to polish them later.

    • at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for

      S:[Σ,X] ∂Σ→ℝ S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}

      a smooth “functional”, namely a smooth map of smooth spaces, its “functional derivative” is simply the plain de Rham differential of smooth functions on smooth spaces

      dS:[Σ,X] ∂Σ→Sℝ→dΩ 1. \mathbf{d}S \colon [\Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.

      The notation can still be optimized. But I am running out of energy now. Has been a long day.

    • (Edited.) An anonymous poster has created a page with Vesselin’s comments on MO simply copied and pasted. I don’t know what others think of this, but whether this is an appropriate use of the nLab seems open to debate. What do others think?

    • I created a page Lax equation having no content so far but soon there will be some content.

    • I just have met Jamie Vicary in Brussels, at QPL 2012. In his nice talk he pointed to an nnLab page which I didn’t know existed:

      It’s about a computer algebra software that can handle KV-2-vector spaces. I have just now added some cross links.

    • I have worked on the general structure of the entry locally presentable category. The previous structure was a bit erratic at times, due to the way it had grown. I have tried to collect paragraphs by topic, give them numbered environments, move theorems from the Examples-section to the Properties-section and so forth.

    • Something odd has been happening at a new entry entitled exchange structure. Someone signing in as Carol entered in quite a lot of material relating to J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. This has just been deleted from the same IP address. This probably means nothing important but it is worth noting.

    • I wanted to be able to point to practical foundations more directly than pointing to foundations and hoping that the reader would spot the paragraph on practical foundations there. So I split off an entry practical foundations. For the moment it contains nothing but the relevant material from foundations copy-and-pasted

    • since I needed to point to it explicitly, not just via the entry BRST complex, I have created an entry ghost field with a paragraph of text.

      I promise that one day I fill in more detailed discussion, but not right now.

    • created metric jet after a mention at the Café.

    • there already was a bit of case distinction at functional between the notion in functional analysis and the nonlinear notion in mapping space theory. I have edited a bit more, trying to polish a bit.

    • Ross Tate has pointed out a mismatch in terminology: Kleisli objects and the Grothendieck construction (of a covariant Cat-valued functor) are both asserted to be “lax colimits”, but they are not the same kind of colimit (the 2-cells go in different directions). Thinking about this more, I have concluded that Kleisli objects are lax colimits and the Grothendieck construction is an oplax colimit. I wrote a bit about my reasoning here. But before I go changing all references to the Grothendieck construction to say “oplax colimit”, I thought I should do a sanity check — does this make sense to everyone else?

    • started complex analytic space

      but I really have some basic questions on this topic, at the time of posting this I am really a layperson:

      is it right that every complex analytic space is locally isomorphic to a polydisk?

      So then they are all locally contractible as topological spaces. Are they also locally contractible as seen by étale homotopy? (So: do they admit covers by polydsisks such that if in the Cech-nerves of these covers all disks are sent to points, the resulting simplicial set is contractible?)

    • I have added some information on the work of Henry Whitehead which is related to this topic, and referred to work of Graham Ellis, and of Higgins and I, which is relevant.

      I expect I have not given the best code for all of this so someone may want to improve it in that respect.

      Graham, also writes in his paper:

      In view of the ease with which Whitehead's methods handle the
      classifications of Olum and Jajodia, it is surprising that the
      papers \cite{olum:1953} and \cite{jaj:1980} (both of which were
      written after the publication of \cite{whjhc:1949}) make
      respectively no use, and so little use, of \cite{whjhc:1949}.

      We note here that B. Schellenberg, who was a student of Olum, has
      rediscovered in \cite{sch:1973} the main classification theorems
      of \cite{whjhc:1949}. The paper \cite{sch:1973} relies heavily on
      earlier work of Olum.

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