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• I created Alex Heller at Jim’s suggestion. It is very stubby and could have a lot more added.

• An old query removed from universal enveloping algebra and archived here:

Eric: Is this a special case of universal enveloping algebra as it pertains to Lie algebras? I thought the concept of a universal enveloping algebra was more general than this. I scribbled some notes here. They are far from rigorous, but the references at the bottom of the page are certainly rigorous. I don’t remember them being confined to Lie algebras. I’m likely confused.

[Edit: Oh! I see now. From enveloping algebra you link to this page and call it enveloping algebra of a Lie algebra. Would that be a better name for this page? Or maybe universal enveloping algebra of a Lie algebra? Something to make it clear this page is specific to Lie algebras?]

Zoran: if you read the above article than you see that it distingusihes the enveloping algebra of a Lie algebra and universal enveloping algebra of a Lie algebra which is a universal one among all such. There is also an enveloping algebra of an associative algebra what is a different notion.

Also added to universal enveloping algebra, a link to a MathOverflow question What is the universal enveloping algebra which is looking for a rather general construction in a class of symmetric monoidal pseudoabelian categories. I also created a minimal literature section.

• New entry (!) tangent Lie algebra. Significant changes at invariant differential form with redirect invariant vector field reflecting the vector fields and other tensor cases. Many more related entries listed at and the whole entry reworked extensively at Lie theory. Some changes at Lie’s three theorems and local Lie group. New stubs Chevalley group and Sigurdur Helgason.

By the way, when writing tangent Lie algebra, I had the problem with finding the correct font for the standard symbol of Lie algebra of vector fields on a manifold. Usually one has varchi symbol which looks like Greek chi but with dash through middle. The varchi symbol is not recognized and I put mathcal X which is slanted and script, just alike, but without dash through middle.

By the way, on a real Lie group $G$ of dimension $n$, if one expresses the right invariant vector field in terms of left invariant vector fields then at each point there is a $\mathbb{R}$-linear operator which sends any frame of left invariant vector fields to the corresponding frame in right invariant vector fields; this gives a $GL_n(\mathbb{R})$-valued real analytic function on $G$ (or, in local coordinates, on a neighborhood of the unit element). In other words, if I take a frame in a Lie algebra and interpret it in two ways, as a frame of left invariant vector fields and a frame in right invariant vector fields, then I can take a matrix of real analytic functions on a Lie group and multiply the frame of left invariant vector fields with this matrix to get the correspoding frame of right invariant vector fields. I use in my current research some computations involving this matrix function. Does anybody know of any reference in literature which does any computations involving this matrix valued function on $G$ ?

• I proudly anounce that nlab has a stubby and shy, but important Hitchin fibration with redirect Hitchin system. Very hot topic nowdays (not only because of Ngo/Laumon proof of the Fundamental Lemma but for many other sides of the story and its importance in physics.

• A stub Massey product and a longer Toda bracket (still plenty gaps of reference, many many unlinked words). No promises w.r.t. spellings.

I now see I’ve missed the convention for capitalization. Will fix that now… done.

Cheers

• I am hereby moving the following old Discussion box from interval object to here

Urs Schreiber: this is really old discussion by now. We might want to start putting dates on discussions. In principle it can be seen from the entry history, but readers glancing at this here hardly will. Maybe discussions like this here are better had at the forum after all.

We had originally started discussing the notion of interval objects at homotopy but then moved it to this entry here. The above entry grew out of the following discussion we had, together with discussion at Trimble n-category.

Urs: Let me chat a bit about what I am looking for here. It seems very useful to have a good notion of what it means in a context like a closed category of fibrant objects to say that path objects are compatibly corepresented.

By this should be meant: there exists an object $I$ such that

• for $B$ any other object, $[I,B]$ is a path object;

• and such that $I$ has some structure and property which makes it “nice”.

In something I am thinking about the main point of $I$ being nice is that it can model compositon: it must be possible to put two intervals end-to-end and get an interval of twice the length. In some private notes here I suggest that:

a “category with interval object” should be

• with a compatible structure of a category of fibrant objects

• and equipped with an internal co-categoy on $\sigma, \tau : pt \to I$ for $I$ the interval object;

• such that $I$ co-represents path objects, in that for all objects $B$, $[I,B]$ is a path object for $B$.

I think there are a bunch of obvious examples: all familiar models of higher groupoids (Kan complexes, $\omega$-groupoids etc.) with the interval object being the obvious cellular interval $\{a \stackrel{\simeq}{\to} c\}$.

I also describe one class of applications which I think this is needed/useful for: recall how Kenneth Brown in section 4 of his article on category of fibrant objects (see theorems recalled there and reference given there) describes fiber bundles in the abstract homotopy theory of a pointed category of fibrant objects. This is pretty restrictive. In order to describe things like $\infty$-vector bundles in an context of enriched homotopy theory one must drop this assumption of the ambient category being pointed. The structure of it being a category with an interval object is just the necessary extra structure to still allow to talk of (principal and associated) fiber bundles in abstract homotopy theory. It seems.

Todd: The original “Trimblean” definition for weak $n$-categories (I called them “flabby” $n$-categories) crucially used the fact that in a nice category $Top$, we have a highly nontrivial $Top$-operad where the components have the form $\hom_{Top}(I, I^{\vee n})$, where $X \vee Y$ here denotes the cospan composite of two bipointed spaces (each seen as a cospan from the one-point space to itself), and the hom here is the internal hom between cospans.

My comment is that the only thing that stops one from generalizing this to general (monoidal closed) model categories is that “usually” $I$ doesn’t seem to be “nice” in your sense here, and so one doesn’t get an interesting (nontrivial) operad when my machine is applied to the interval object. But I’m generally on the lookout for this sort of thing, and would be very interested in hearing from others if they have interesting examples of this.

to be continued in the next comment

• I noticed that the text at loop space didn’t point to smooth loop space and didn’t make clear that such a variant might even exist. So I have now expanded the Idea-section there a little to give a better picture.

• I have received a question on the old entry directed object, so I am looking at that now. First of all I’ll clean it up a bit and move old discussion from there to here:

[begin forwarded discussion]

+–{.query}

Eric: I don’t fully “grok” this constructive definition, but I like its flavor. Is it possible to formalize the procedure in a simple catchy phrase? In other words, when you begin with a “category $C$ with interval object $I$”, but whose objects are otherwise undirected (like Top), you construct the “supercategory $d_I C$” with directed $C$-objected (even though no objects in $C$ are directed). I used the term “directed internalization”, but is there a better term?

I just think this concept is important and should have some really slick arrow theoretic description and I’m not having any luck coming up with one myself.

=–

[ continued in next post ]

• finally expanded the long-existing table of contents complex geometry - contents and included it as a floating TOC in the relevant entries.

Do we have more entries that need to go here and which I have forgotten?

• did I say that I created Theta space?

This is a really nice model. Rezk claims to have shown to get the homotopy hypothesis right for all (n,r)-categories and for both n and r ranging to $\infty$ . If that holds water, it's quite impressive. It seems the only thing missing then is the $(n+1,k+1)$- Theta-space of all $(n,k)$-Theta spaces. Does anyone know if there is a proposal for that?

It's also interesting how the result is a mix of globular and simplicial shapes. So in what respect does that build on/improve over Joyal's original proposal?

• A query about the new entry on copncurrency theory: Does ‘simultaneously’ make sense if there is no global clock?

If not, then the situation gets a lot more like some models for spacetime and the idea of slices through some evolving state space might be a good model.

• Someone, apparently in Berlin, has created a page called www.mfo.de/document/1145/OWR_2011_52.pdf, with just that text (and ’My First Slide’) in the body. The URL points to a report on a logic workshop at Oberwolfach around this time last year. It’s not spam, but what should we do with it?

• Someone signing themselves as ‘Joker? at November 3, 2012 08:05:13 from 93.129.88.58’ deleted two lines from sheaf and topos theory. There seemed no reason for this, so I have rolled back to the previous version.

• The recent changes to the various modal logic pages have changed the emphasis from the ’many agent’ versions $S4(m)$.etc. to a type theoretic one. That would be okay but in so doing they have become a bit garbled so they refer to K(m) but then just describe $K$ itself. I am wondering what is planned for these. I originally wrote them with the aim of increasing the nPOV side of the Computer Science entries and to have some brief introduction to modal logs, what should they become?

• October 24, 2012 09:26:08 by Anonymous Coward (99.133.144.164) has added a comment questioning the validity of a sentence at reflective subcategory.

• wanted to be able to say sum and have a pointer to somewhere.

• I made starts on lexicographic order and on compactification. Lexicographic order was defined only for products of well-ordered families of linear orders (probably the most common type of application).

I’m not very happy with the opening of compactification.

• I edited the old entry projection a little.

There is no real systematics in common use of “projection” as opposed to “projector”, but I think the following makes good sense:

1. a projection is a canonical map out of a product;

2. a projector is an idempotent in a suitably abelian category

and then the relation is: A projector is a projection followed by a subobject inclusion.

That’s how I have now put it in the entry.

• New entry enumerative geometry. New stubs Schubert calculus, intersection theory.

By the way (Andrew); the title of this nForum post is not seen but truncated. This happens because of some other stuff is placed into the corner in the same line. It says unimportant info “Bottom of Page” preceded by long space between the truncated title and this info ad. I think it is more important that the titles be spelled entirely.

• I had need to give idempotent a bit more of expository text. So I wrote a bit in the Idea-section. This now necessarily overlaps with what follows. Deserves to be edited further.

• created a table of contents idempotents - contents and included it as a floating TOC into the relevant entries

• While writing at k-morphism, I noticed there is no article on globular operad (aka Batanin operad), so I wrote one. Experts please look over, and improve if desired.

• While writing the new Idea-section now at Segal condition I felt the need to have a table of contents

So I started one and added it to the relevant entries as a floating TOC.

• I think we need a floating table of contents categories of categories - contents to connect our entries on related topics. I have started one.

But this needs to be further expanded. also haven’t included it into the relevant entries yet, no time right now.

• I have written out in some detail the proof at Grothendieck spectral sequence.

But I still need to go through it and proof-read and polish. Handle with care for the moment. Maybe the whole thing needs to be rearranged, for readability.

• have now spelled out at Tor in simple terms how $Tor_1^{Ab}(A,B)$ is a torsion group, so far for the case that $A$ is finite.

• added to the definiton-section at geometric category a pointer to the alternative terms “logical category” and “pre-logos”.

• I (only) now realize that I pretty much missed the story of familial regularity and exactness. But also it was easy to miss, with the entries that are unified by this not pointing back to it.

To rectify this I have created now a floating TOC and am including it into all the relevant entries:

Please check out that TOC and edit/modify as need be.

• The reference provided today on the CatTheory mailing list

• A.R. Garzón, J.G. Miranda, Serre homotopy theory in subcategories of simplicial groups Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123

I have added to k-tuply groupal n-groupoid, and also to n-group and infinity-group

• I've done a tiny bit of work to add a more intuitive introduction to the concepts of model category and Quillen equivalence, and I plan to do some more. If anyone wants to help, that would be great. For example, it would be nice to give some general intuition for fibrations, cofibrations and weak equivalences and why they matter.
• I need to point to reduced homology, so I created an entry. But nothing much there yet.

• I did some editing over at free module, under the section on submodules of free modules. I don’t have Rotman’s book before me, so I can’t check whether he assumes the commutativity hypothesis for proposition 2, but I put it in to be safe. (Actually, I’ll bet it’s needed, since we have to be careful around invariant basis number which holds for commutative rings.) The proof that I added does use this hypothesis.

Also, I deleted the remark that this is the Nielsen-Schreier theorem in the case $R = \mathbb{Z}$, since NS refers to groups as opposed to abelian groups.

• created an entry titled Topological Quantum Field Theories from Compact Lie Groups

on the recent (or not so recent anymore) article by Freed-Hopkins-Lurie-Teleman (therefore the capizalization).

I typed into this a summary of their central proposal for how to formalized the path integral quantization for "direcrete" quantum field theories, in terms of higher category theory.

I think this is important, and is actually a simple idea, but few people having looked at the article maybe get away with the take-home message here. So I tried to amplify this.

I also have some own thoughts about this. So I put a big query box in the end, with a question.