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started infinity-Lie algebroid valued differential forms , since that is needed all through our discussion of oo-Chern-Weil elsewhere. But right now the entry is stubby.
I wrote down a definition at pseudofunctor.
created Chern-Simons circle 7-bundle
created stubs classical Lie group and exceptional Lie group and linked to them from simple Lie group (all very stubby)
New article: opposite magma (including monoids, groups, rings, algebras, etc).
without really intending to do so and certainly without having the time to do so, I ended up creating some stubs for
following a public demand, I have
created inner derivation Lie 2-algebra,
cross-linked it with Cartan calculus and pointed out how they are related,
and started at Weil algebra a section As the CE-algebra of the L-oo algebra of inner derivations.
expanded algebraic Kan complex. Added Idea-section and Properties-section
When I told him about it today, Dmitry Roytenberg urged that we highlight a bit more visibly a fact that was mentioned on the nLab before, but not highlighted maybe sufficiently.
Namely by the general theory of infinity-Lie algebra cohomology we have for every -Lie algebroid equipped with an invariant polynomial the corresponding Chern-Simons wich exhibits the transgression to the corresponding -Lie algebroid cocycle.
If you apply this general theory to a Poisson Lie algebroid, then then Chern-Simons form that drops out is the action functional of the Poisson sigma-model.
I added this remark more visibly now to Poisson Lie algebroid, Poisson sigma-model and infinity-Lie algebroid cohomology.
Following sections 34 and 35 in Dwyer-Hirschhorn-Kan-Smith (DHKS), I have begun to write up a page on hammock localization and simplicial Grothendieck construction localization. simplicial localization of a homotopical category.
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.
For some bizarre reason I added a lot of information to the page Pontrjagin duality.
created coherence law
(was surprised to find that we didn’t have this already. Or do we?)
I clarified how this works in single-sorted definition of category
created stubs for (2,1)-presheaf and hom-groupoid, just because I needed to link to them
I separated binary digit from boolean domain.
the entry geometric definition of higher categories had been in a sorry state. I have edited it a bit in an attempt to bring it into shape
stub for 3-morphism
created infinity-anafunctor
created (or edited)
Here the first item is always defined as a coycle in, respectively, the
And I tried to establish the same kind of link pattern for
created Poisson Lie algebroid
New entries ribbon graph, mapping class group, Jakob Nielsen and just one reference at Mumford conjecture. Mainly to set some background for yesterday started entries graph homology, noncommutative symplectic geometry and related effort to understanding of the characteristic classes for infinity algebras in work of Andrey Lazarev and Alastair Hamilton who use variants of the Kontsevich’s construction involving graph (co)homology.
created vertical composition
Several new articles have been announced on the thread Entropy, and many don’t have much to do with that subject. So I’ll just list them all here. (Many of them already have discussions on the other thread, however, so do look there.)
stub for nonabelian bundle gerbe, since I need the link at infinity-Weil theory introduction
started a stub for pseudo-connection, in order to satisfy links. But now I am really too tired. More tomorrow.
New entry formal noncommutative symplectic geometry within the circle of entries related to graph homology.
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.
I expanded linear algebra and wrote linear operator.
started semisimple Lie algebra
trivial group, trivial ring, trivial Lie algebra, abelian Lie algebra (which may not be trivial exactly but by special dispensation is still too trival to be simple)
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.
There is now a decvent Idea-section;
I created three subsections for three different kinds of constructions of this beast.
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!
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.
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.
@Urs: I answered your query over at homotopy category of an (infinity,1)-category. The Ho(SSet) enrichment is given by the lax monoidal functor SSet -> Ho(SSet).
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.
There was more about monoid objects at monoid than at monoid object, so I incorporated the latter into the former. (This means that the history of the latter is now at monoid object > history.
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.
Stubs at composable pair and commutative triangle
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.
started one-line stubs for matrix Lie group and matrix Lie algebra just so that the links to these, which started appearing at parallel transport and elsewhere do point somewhere
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 a field this is an empty condition and we are in business and find the familiar model structures. But for not a field? Notably simply 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 fixed in the first coordinate, , where is the canonical inclusion, is a functor . Then the undercategory construction gives the adjoint to this functor sending . This makes it substantially clearer to understand what’s going on, since is the set of those maps such that .
Lurie’s notation is nonstandard and inferior, since it obscures the obvious adjointness property.
The definition for overcategories is “dual” (by looking at the join of on the right).
started nice simplicial topological space with material provided by David Roberts
stub for simplicial topological space
considerably expanded Lie infinity-groupoid. But still stubby.
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?
I linked to it, so here is jointly epimorphic family, our newest stub.
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).