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    • I could have sworn that we had something for thin category, at least a redirect, but we don’t. Or didn’t. Now we do.

      Not much to it, just a note of terminology, like inductive limit or (0,1)-topos.

      There’s also a diagram that I can’t to get to work there, if anybody wants to help.

    • This is mathematically much simpler than the classical Gleason’s Theorem, but I added it to Gleason’s theorem anyway.

    • In gluing categories from localizations (zoranskoda) the main section

      From a family of localizations to a comonad

      is fully rewritten in improved notation. In other way, it is explained better how to get a comonad from a cover of a category by not necessarily compatible flat localizations. This generalizes the Sweedler's coring to relative situations. Now from such data one can make a two category, which I will explain in few days.

      This is a preliminary to something I am writing at the moment namely to explain in such terms actions of comonads and monoidal categories on such descent categories. This part will be analogous to description of equivariant maps among G-manifolds in pairs of local charts, but because of the distributive laws with coherences, the thing complicates.

    • I started quantifier, but I ran out of time to say all that I wanted. I’ll probably get back to it in a couple of hours.

    • I redid everything that includes contents using the new click-based menu system. This includes HomePage; there didn’t seem to be a need anymore to have two columns, so I put them back in one column. However, those are separate issues; we could put them back in two columns again and still do the click thingy.

    • I added some stuff about states in statistical physics to state.

    • A couple of new pages have appeared, theory of primes and PrimeDeGold, the latter being the author of the former (and itself). They seem like nonsense, but perhaps someone (Andrew? Toby?) is doing testing? Or else testing for a sort of spamming?

    • I added word "abelian" when the article Lie algebra cohomology talks about Lie algebra cohomology as cohomology of CE complex. While one can combine this abelian cohomology with various actions to get nonabelian information, there is also a genuine nonabelian cohomology of Lie algebra as it is for Lie groups. At least some special cases are understood in low dimensions. For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of factor systems. I will create some entry about that, and put a longer exposition in "private" web following my old notes from Spring/Summer 1997. For start I just started a stub nonabelian Lie algebra cohomology.

    • created and expanded infinity-Lie algebra cohomology.

      There is now a section on the (∞,1)(\infty,1)-topos theoretic interpretation, and one on how to understand ∞\infty-Lie algebra extensions as special cases of the general nonsense on principal ∞\infty-bundles.

      The discussion leaves quite a bit of room for polishing, but I don’t feel like spending much more time on this right this moment.

    • shrinkable map, so I can reference it for Urs’ question on Hurewicz fibrations.

    • I wrote one following the pattern set by zero.

      Actually, it’s not terribly lonely; three pages link to it!

    • I added new links to my nLab page. I have links to my ’table of categories’ and ’database of categories’ here

    • a description of a free monad on Sets in Cat, the successor monad, and remarks.

    • In my personal nnlab a bibliography for an interesting topic, which some of my peers in Zagreb got recently interested in: Feynman proof of the Lorentz force equations (zoranskoda). It is funny – deriving gauge theories and even gravity just from commutation relations for the generators (coordinates and “covariant momenta”) without any action principle, that is without assuming Lagrange or Hamilton formalism to hold. It has nice extensions and it may be important for the philosophy of gauge theories aka connections on vector bundles. Any ideas of categorification may be interesting…

      When talking personal nnlab I wrote a long general advice page for students who may ask for my future mentorship in Zagreb of some sort. Well, if I stay in Zagreb. The things are getting rough for science here, and I am not sure of my own future.

    • Hisham Sati in January posted a survey Geometric and topological structures related to M-branes. In a hard effort of several hours of intense work I created an entry containing hyperlinked bibliography of that article (I took LaTeX source, scraped off various LaTeX commands like bf, it, bibitem etc. and then started creating various hyperlinks). Most of the hyperlinks to the arxiv and few to the project euclid are created so far. Many items still do not have proper external links which would be very welcome. This is a very nice bibliography for something of much interest to Urs, me and some other nlabizants, and I would like to have it practical for our systematic online study.

    • Added a proof of the pasting lemma to pullback, and the corresponding lemma to comma object (also added the construction by pullbacks and cotensors there).

    • I have added a stub entry to the lab on Dominique Bourn. There are quite a few links that need developing there as the protomodular category stuff is quite rudimentary. I would need to learn more about it to fill things up so if anyone does feel they can help, please charge ahead.

    • On variety of algebras appears the sentence “(This paragraph may be original research. Probably the concept does appear in the literature but under a different name.)”. The paragraph in question is about typed varieties of algebras. Looking at the history, this sentence (and indeed, the whole page!) appears to be due to Toby (Bartels).

      I’m curious as to what part that sentence refers to, in particular due to my interest in what I call graded varieties of algebras (nomenclature coming from algebraic topology and graded cohomology theories), which I thought was just an example of a heterogeneous variety of algebras, a term that I’ve come across in the literature. Certainly the concepts feel closely related, and it took a fair amount of paper chasing to find the term “heterogeneous” (though “many-sorted” theories seemed a bit more of a common term), but despite my interest, I’m no expert and am sure I’m missing something. Problem is: I don’t know what and I don’t know how to properly formulate my question!

      (Added in edit): Actually, I see that the term “multisorted” is in use on Lawvere theory.

    • The definition of “subnet” under net looked wrong to me (part of the wrongness was obvious), so I changed it so that it looks correct to me. Could someone please give independent verification?

    • A semester ago I announced a possible mentorship in Zagreb if a physics student would like to take to digest and write a diploma on the basis of Baez-Schreiber work on higher gauge theory. Nobody chose the topic but the page in my personal nlab is left out from those times, and maybe it will be recycled by a future announcement, though it is questionable as I am likely to leave my present institution in few months. But in order to be functional, it is good to have also the list of literature which i just compiled, including the appended list of very advanced references so that it might serve at all levels. Suggestions and usage for your own purposes are welcome.

      Yes: diplomski higher gauge theories (zoranskoda)

    • added a stub entry for holonomy.

      Just the bare definition, and of that even only the most naive one. Don’t have time for more. But created it anyway because I needed the link.

      (Sounds a bit like like: I was young and needed the money…)

    • We talk of a ’homogeneous linear functor’ at Goodwillie calculus, a functor which maps homotopy pushout squares to homotopy pullback squares. There are also higher degree homogeneous functors which map (n+1)(n+1)-dimensional cubical homotopy pushout diagrams to (n+1)(n+1)-dimensional cubical homotopy pullback diagrams. These allow polynomial approximation in the functor calculus.

      We also have linear functor and polynomial functor. I take it that these latter two are unrelated to each other, and to the functor calculus terms. I think we need some disambiguation.

      Does anyone know why in the Goodwillie calculus those functors are called linear? Perhaps this helps:

      At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.

      A simple and familiar example of this is the Euler characteristic e(X)e(X), where the local-to-global property for good decompositions takes the form e(U∪V)=e(U)+e(V)−e(U∩V)e(U \union V) = e(U) + e(V) - e(U \cap V). A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer–Vietoris sequence. Finally one can consider the functor SP ∞:Top→TopS P^{\infty}: Top \to Top, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups π *(SP ∞(X))=H *(X)\pi_*(SP^{\infty}(X)) = H_*(X), the Meyer-Vietoris sequence for homology is thus a consequence of applying π *(−)\pi_*(-) to the homotopy pullback square.

      It was the insight of Tom Goodwillie in the 1980’s that such “linear” functors F:Top→TopF: Top \to Top form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree nn takes appropriate sorts of (n+1)(n+1)-dimensional cubical homotopy pushout diagrams to (n+1)(n+1)-dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

    • I am a bit stuck/puzzled with the following. Maybe you have an idea:

      I have a group object GG and a morphism G→QG \to Q. I have a model for the universal GG-bundle EG\mathbf{E}G (an object weakly equivalent to the point with a fibration EG→BG\mathbf{E}G \to \mathbf{B}G).

      I have another object EQ\mathbf{E}Q weakly equivalent to the point such that I get a commuting diagram

      G → Q ↓ ↓ EG → EQ \array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q }

      Here QQ is not groupal and i write EQ\mathbf{E}Q only for the heck of it and to indicate that this is contractible and the vertical morphisms above are monic (cofibrations if due care is taken).

      So I have GG acting on EG\mathbf{E}G and the coequalizer of that action exists and is BG\mathbf{B}G

      G×EG→→EG→BG G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{B}G

      I can also consider the colimit KK of the diagram

      G×EG→→EG→EQ. G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{E}Q \,.

      That gives me a canonical morphism BG→K\mathbf{B}G \to K fitting in total into a diagram

      G → Q ↓ ↓ EG → EQ ↓ ↓ BG → K. \array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& K } \,.

      Now here comes finally the question: I know that the coequalizer of G×EG→→EGG \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G is a model for the homotopy colimit over the diagram

      ⋯G×G→→→G→→* \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *

      as you can imagine. But I am stuck: what intrinsic (∞,1)(\infty,1)-categorical operation is KK a model of?

      I must be being dense….

    • In the article categorification via groupoid schemes, I removed a distracting query box containing a discussion of how to get a double slash in TeX. The answer was that // works, but is ugly, while prettier things like \sslash may not work for people who don't have the font loaded.

    • added a stubby

      to the entry Lie infinity-groupoid.

      The punchline is that if we pick a groupal model for EG\mathbf{E}G – our favorite one is the Lie 2-group INN(G)INN(G) – then by the general nonsense of Maurer-Cartan forms on ∞\infty-Lie groups there is a Maurer-Cartan form on EG\mathbf{E}G. This is, I claim, the universal Ehresmann connection on EG\mathbf{E}G.

      The key steps are indicated in the section now, but not exposed nicely. I expect this is pretty unreadable for the moment and I tried to mark it clearly as being “under construction”. But tomorrow I hope to polish it .

    • created topological submersion. I’ve seen more than one definition of this, and both could be useful. My natural inclination is to the more general, where each point in the domain has a local section through it.

      On a side note I use a related condition in my thesis for a topological groupoid over a space: every object is isomorphic to one in the image of a local section. This was used in conjunction with local triviality of topological bigroupoids to define certain sorts of 2-bundles.

    • expanding the entry hypercohomology started by Kevin Lin, I wrote an Idea-section that tries to explain the nnPOV on this

    • Edited Lie groupoid a little, and new page: locally trivial category. There is an unsaturated link at the former, to Ehresmann’s notion of internal category, which is different to the default (Grothendieck’s, I believe). The difference only shows up when the ambient category doesn’t have all pullbacks (like Diff, which was Ehresmann’s pretty much default arena). It uses sketches, or something like them. There the object of composable arrows is given as part of the data. I suppose the details don’t make too much difference, but for Lie groupoids, it means that no assumption about source and target maps being submersions.

      The latter page is under construction, and extends Ehresmann’s notion of locally trivial category/groupoid to more general concrete sites. I presume his theorem about transitive locally trivial groupoids and principal bundles goes through, it’s pretty well written.

    • created Bianchi identity.

      (gave it the ∞\infty-Lie theory toc, but already with the new CSS code. So as soon as that CSS code is activated on the main nnLab, that TOC will hide itself and become a drop-down menu. I think.)

    • Were we to have an entry on the cosmic cube, would people be happy with that name, or should we have something less dramatic?

    • I’ve added some items to mathematicscontents.

      I never did much with the contents pages, so I may not have organised this in the best way.

    • You can turn a set into a topological abelian group by equipping it with a family of G-pseudonorms.

    • Does anyone have any notes, or know of anyone who has notes, from Igor’s Oberwolfach or Utrecht talks?

    • created 2-site with the material from Mike’s web (as he suggested). Added pointers to original articles by Ross Street.

    • I started a stub on plethysm.

      Does anyone know how this mathematical term originated? I hear someone suggested it to Littlewood. But who? And why? And what’s the etymology, exactly?

    • I have a query for Mike, or anyone who wants to tackle it, over at locally finitely presentable category. Mike seems to be saying that only the category of models of a finitary essentially algebraic theory is locally finitely presentable, but some paper seems to suggest otherwise...

    • I’m struggling to further develop the page on Schur functors, which Todd and I were building. But so far I’ve only done a tiny bit of polishing. I deleted the discussion Todd and I were having near the top of the page, replacing it by a short warning that the definition of Schur functors given here needs to be checked to see if it matches the standard one. I created a page on linear functor and a page on tensor power, so people could learn what those are. And, I wound up spending a lot of time polishing the page on exterior algebra. I would like to do the same thing for tensor algebra and symmetric algebra, but I got worn out.

      In that page, I switched Alt to Λ\Lambda as the default notation for exterior algebra. I hope that’s okay. I think it would be nice to be consistent, and I think Λ\Lambda is most widely used. Some people prefer ⋀\bigwedge.

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