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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.
am starting curvature characteristic form and Chern-Simons form.
But still working…
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).
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.
created circle n-bundle with connection.
See the nForum thread on oo-Chern Weil theory for background.
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.
started stacked cover
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.
I moved states in AQFT and operator algebra to state on an operator algebra, and I also included there a redirect from state on an algebra, which had been requested on quantum state. Hopefully it makes sense that way.
I almost wrote state on an algebra myself, but I’m glad that I didn’t, since everything that I would have said (and more) was already on the extant page.
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?
created and expanded infinity-Lie algebra cohomology.
There is now a section on the -topos theoretic interpretation, and one on how to understand -Lie algebra extensions as special cases of the general nonsense on principal -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 added new links to my nLab page. I have links to my ’table of categories’ and ’database of categories’ here
Baues-Wirsching cohomology of small categories got a stub. Updates to Hans-Joachim Baues.
New entry Masoud Khalkhali related to the quite recent cyclic cohomology entry (which still lacks the basic material).
In my personal 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 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’ve started infinitary Lawvere theory following this n-Forum discussion.
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
and am heading for
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?
I found a good constructive definition of proper subset and put it in there. Also I wrote improper subset.
Edit: also family of subsets; see below.
started floating TOC differential cohomology - contents
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.
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…)