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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 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 $R$ a field this is an empty condition and we are in business and find the familiar model structures. But for $R$ not a field? Notably simply $R = \mathbb{Z}$ 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 $K$ fixed in the first coordinate, $S\mapsto i_K^{K\star S}: K\to K\star S$, where $i_K^{K\star S}$ is the canonical inclusion, is a functor $SSet\to (K\downarrow SSet)$. Then the undercategory construction gives the adjoint to this functor sending $(K\downarrow SSet) \to SSet$. This makes it substantially clearer to understand what’s going on, since $Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F)$ is the set of those maps $f:K\star S\to X$ such that $f|K=f\circ i_K^{K\star S}=F$.

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

The definition for overcategories is “dual” (by looking at the join of $K$ 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.

• 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 $(\infty,1)$-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 $n$lab 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 $n$lab 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.

• am starting to create stubs