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    • New article: opposite magma (including monoids, groups, rings, algebras, etc).

    • At period it was claimed that the ring of periods P⊂ℂP\subset \mathbb{C} is a subfield of ℂ\mathbb{C}. It is conjectured (see, e.g. wikipedia) that 1/π1/\pi is not a period, and since π\pi is a period, PP is not expected to be a field. I’ve fixed this up.

    • Mike has added to Georges Maltsiniotis a link to an English version of Maltsiniotis’ version of Grothendieck’s version of an oo-groupoid. I can’t believe I missed this on the arXiv yesterday! The French version (or rather, the French predecessor) of this paper is one of those things I wish I could sit down and read in detail and pull apart, but haven’t the time (and the language was a small, but psychological barrier).

    • Created stub homotopy algebra as this is traditional terminology unifying L, A, G, B etc infinity algebras for references and disambiguation and for operadic point of view.

    • created coherence law

      (was surprised to find that we didn’t have this already. Or do we?)

    • Growing out of my recent edit to the anafunctor article, I've created a new article on cliques.

    • noticed that the entry curvature was in all its stubiness already a mess.

      So I tried to write an Idea-section that indicates how the notion of curvature appears for embedded surfaces and then gradually generalizes to that of connections on bundles and further.

      Eventually I would like to split off the section on extrinsic curvature to a separate entry extrinsic curvature and Gaussian curvature.

      But not now, I need to be doind something else…

    • started a stub for pseudo-connection, in order to satisfy links. But now I am really too tired. More tomorrow.

    • I added some more variations, links, and references to string diagram. I’m sure there are a lot more references that ought to go there.

    • there is a bunch of things on my to-do list concerning write-up of stuff on the nLab in the big context of my “diff cohomology in an (oo,1)-topos”-writeup. I am lagging behind. I could use some more help!

      Of course you all are busy with your own stuff. But maybe there is a tiny chance somebody reads this here, maybe somebody who has been lurking all along, somebody who would enjoy helping out. I could say: I offer 60 reputation points! More seriously, this might be a chance to dip your feet into the water and join in to some interesting research. Here is the deal:

      I have a LaTeX writeup of a fairly long proof that establishes the weak equivalence of 3 different strict 2-group models of the string 2-group. It appears as the proof of prop. 5.25 on p. 94 of these notes here. The trouble (for me) is that this proof involves some diagrams that would require code-cogs for implementation on the nLab. I want these diagrams on the nLab!

      I have the LaTeX source code, of course, so it’s not much work to make this run on the nLab! But a bit of work. A tad more work than I find leisure for right now.

      Let me know if you are interested and I’ll send you the LaTeX source!

      Best, Urs

    • I considerably expanded and reorganized the discussion at Chern-Simons 2-gerbe.

      1. There is now a decvent Idea-section;

      2. I created three subsections for three different kinds of constructions of this beast.

      3. The first subsection contains now a detailed account of the consztruction by Brylinski-McLaughline of an explicit Cech-Deligne cocycle. I end this with saying that once the construction is there, proving that it is correct is easy. The mystery is maybe how one comes up with the construction in the first place!

      4. So then in the next subsection I rederive this construction as a special case of the general methods described at infinity-Chern-Weil theory. So I show that from Lie integration of the underlying Lie algebra cocycle one gets a canonical lift to pseudo-connections with values in the Lie algebra, and turning the crank, out drops the Brylinski-McLaughlin construction. I’ll later see if I can streamline this discussion a bit more.

      5. Then there is a third subsection which is supposed to deal with the construction of bundle 2-gerbe representatives. But here I am being lazy and just give the references so far. Even though the construction is actually simple.

    • Using codecogs recipe and ascii table I wrote short entries fork and split equalizer. For those who distinguish fork and cofork, I have hard time remembering which one is which one.

      By the way, nForum is today having lots of problems on my computer, it asks for human recognition, it bails out my automatically remembered password many times out and resets the settings for markdown when writing etc. often.

    • I rearranged the template page so that the template came up top, on the grounds that this is what people will mostly want to copy and paste. Then they can scroll down for a more detailed example.

    • I've updated Contributors for this month. If there are any mistakes, I won't find them until October.

    • The definition at simple object referred to subobjects instead of quotient objects. Although these definitions are equivalent in abelian categories, it seems to me that we must use quotient objects to get the correct definition of a simple group, so I have changed it.

    • I have added a bit of history to the entry on Baues-Wirsching cohomology. Whilst looking for something else I found a paper by Charlie Wells from 1979, extending the earlier ideas of Leech cohomology for semigroups to small categories. He defines various types of extension and classifies them using the same methods as B and W used a few years later.

    • Added a bit to skeleton about skeletons of internal categories

    • added to exact functor a new subsection “Between abelian categories” and listed there (briefly) the standard characterizations of left/right exact functors in terms of preservation of left/right exact sequences.

      Also added a reference by Michael Barr on the relation between exactness and respect for homology in very general contexts.

    • added to injective object propositions and examples for injective modules and injective abelian groups

      P.S. I am checking if I am missing something: Toën on page 48/49 here behaves as if it were clear that there is a model structure on positive cochain complexes of R-modules for all R in which the fibrations are the epis. But from the statements that i am aware of at model structure on chain complexes, in general the fibrations may be taken to be those epis that have injective kernels. For RR a field this is an empty condition and we are in business and find the familiar model structures. But for RR not a field? Notably simply R=ℤR = \mathbb{Z} What am I missing?

    • I fixed the definition at over quasi-category so it makes the adjointness relationship clearer between overcategories and joins. In particular, Lurie’s notation and definition makes it very hard to see this. It’s much easier to see what’s going on when we look at things as follows: The join with KK fixed in the first coordinate, S↦i K K⋆S:K→K⋆SS\mapsto i_K^{K\star S}: K\to K\star S, where i K K⋆Si_K^{K\star S} is the canonical inclusion, is a functor SSet→(K↓SSet)SSet\to (K\downarrow SSet). Then the undercategory construction gives the adjoint to this functor sending (K↓SSet)→SSet(K\downarrow SSet) \to SSet. This makes it substantially clearer to understand what’s going on, since Hom SSet(S,X F/):=Hom K↓SSet(i K K⋆S,F)Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F) is the set of those maps f:K⋆S→Xf:K\star S\to X such that f|K=f∘i K K⋆S=Ff|K=f\circ i_K^{K\star S}=F.

      Lurie’s notation Hom SSet(S,X F/):=Hom F(K⋆S,X)Hom_{SSet}(S, X_{F/}):= Hom_F(K\star S, X) is nonstandard and inferior, since it obscures the obvious adjointness property.

      The definition for overcategories is “dual” (by looking at the join of KK on the right).

    • Aleks Kissinger has given us sifted colimit. Although I don’t quite understand the definition.

    • I revisited some old discussion with Mike at sequence. Are you happy now, Mike?

    • started oo-vector bundle on my personal web, following my latest remarks in the thread here on deformation theory.

    • New page: indecomposable object, following (what I think is) Johnstone's definition. I also found it in some online topos theory lecture notes by Ieke Moerdijk and Jaap van Oosten.

      Lambek and Scott give a different definition in Intro. to Higher-order Cat. Log., p. 168. I'm not sure how it relates to Johnstone's.

      I've also given a proof that indecomposable <=> connected in an extensive category. I'd be interested to know whether this hypothesis is the weakest possible, if anyone has any ideas (or just likely-looking references).

    • 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.

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