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

• 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

• 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?

• 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…)

• We talk of a ’homogeneous linear functor’ at Goodwillie calculus, a functor which maps homotopy pushout squares to homotopy pullback squares. There are also higher degree homogeneous functors which map $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. These allow polynomial approximation in the functor calculus.

We also have linear functor and polynomial functor. I take it that these latter two are unrelated to each other, and to the functor calculus terms. I think we need some disambiguation.

Does anyone know why in the Goodwillie calculus those functors are called linear? Perhaps this helps:

At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.

A simple and familiar example of this is the Euler characteristic $e(X)$, where the local-to-global property for good decompositions takes the form $e(U \union V) = e(U) + e(V) - e(U \cap V)$. A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer–Vietoris sequence. Finally one can consider the functor $S P^{\infty}: Top \to Top$, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups $\pi_*(SP^{\infty}(X)) = H_*(X)$, the Meyer-Vietoris sequence for homology is thus a consequence of applying $\pi_*(-)$ to the homotopy pullback square.

It was the insight of Tom Goodwillie in the 1980’s that such “linear” functors $F: Top \to Top$ form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree $n$ takes appropriate sorts of $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

• I am a bit stuck/puzzled with the following. Maybe you have an idea:

I have a group object $G$ and a morphism $G \to Q$. I have a model for the universal $G$-bundle $\mathbf{E}G$ (an object weakly equivalent to the point with a fibration $\mathbf{E}G \to \mathbf{B}G$).

I have another object $\mathbf{E}Q$ weakly equivalent to the point such that I get a commuting diagram

$\array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q }$

Here $Q$ is not groupal and i write $\mathbf{E}Q$ only for the heck of it and to indicate that this is contractible and the vertical morphisms above are monic (cofibrations if due care is taken).

So I have $G$ acting on $\mathbf{E}G$ and the coequalizer of that action exists and is $\mathbf{B}G$

$G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{B}G$

I can also consider the colimit $K$ of the diagram

$G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{E}Q \,.$

That gives me a canonical morphism $\mathbf{B}G \to K$ fitting in total into a diagram

$\array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& K } \,.$

Now here comes finally the question: I know that the coequalizer of $G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G$ is a model for the homotopy colimit over the diagram

$\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *$

as you can imagine. But I am stuck: what intrinsic $(\infty,1)$-categorical operation is $K$ a model of?

I must be being dense….

• In the article categorification via groupoid schemes, I removed a distracting query box containing a discussion of how to get a double slash in TeX. The answer was that // works, but is ugly, while prettier things like \sslash may not work for people who don't have the font loaded.

to the entry Lie infinity-groupoid.

The punchline is that if we pick a groupal model for $\mathbf{E}G$ – our favorite one is the Lie 2-group $INN(G)$ – then by the general nonsense of Maurer-Cartan forms on $\infty$-Lie groups there is a Maurer-Cartan form on $\mathbf{E}G$. This is, I claim, the universal Ehresmann connection on $\mathbf{E}G$.

The key steps are indicated in the section now, but not exposed nicely. I expect this is pretty unreadable for the moment and I tried to mark it clearly as being “under construction”. But tomorrow I hope to polish it .

• created topological submersion. I’ve seen more than one definition of this, and both could be useful. My natural inclination is to the more general, where each point in the domain has a local section through it.

On a side note I use a related condition in my thesis for a topological groupoid over a space: every object is isomorphic to one in the image of a local section. This was used in conjunction with local triviality of topological bigroupoids to define certain sorts of 2-bundles.

• expanding the entry hypercohomology started by Kevin Lin, I wrote an Idea-section that tries to explain the $n$POV on this

• Edited Lie groupoid a little, and new page: locally trivial category. There is an unsaturated link at the former, to Ehresmann’s notion of internal category, which is different to the default (Grothendieck’s, I believe). The difference only shows up when the ambient category doesn’t have all pullbacks (like Diff, which was Ehresmann’s pretty much default arena). It uses sketches, or something like them. There the object of composable arrows is given as part of the data. I suppose the details don’t make too much difference, but for Lie groupoids, it means that no assumption about source and target maps being submersions.

The latter page is under construction, and extends Ehresmann’s notion of locally trivial category/groupoid to more general concrete sites. I presume his theorem about transitive locally trivial groupoids and principal bundles goes through, it’s pretty well written.

• created Bianchi identity.

(gave it the $\infty$-Lie theory toc, but already with the new CSS code. So as soon as that CSS code is activated on the main $n$Lab, that TOC will hide itself and become a drop-down menu. I think.)

• I created hypermonoid, polishing the comments I made in the hypermonoid thread into an article. The last subsection of the article mentions a general technique for constructing hypermonoids which ought to immediately suggest further examples to a quantum group specialist like Zoran, but I am not such a specialist. I also inserted some shameless self-promotion under References.

• Were we to have an entry on the cosmic cube, would people be happy with that name, or should we have something less dramatic?

• I worked on Nonabelian Algebraic Topology

• made the entry “category: reference”. all about the book by Brown et al – if we feel we need a more generic entry with lower case title later, we can still split it off again

• then I started adding a “Contents” section similar to what we have at Elephant and Higher Topos Theory etc., and started adding some of the content of relevance for the cosmic cube.

• I’ve added some items to mathematicscontents.

I never did much with the contents pages, so I may not have organised this in the best way.

• You can turn a set into a topological abelian group by equipping it with a family of G-pseudonorms.

• Does anyone have any notes, or know of anyone who has notes, from Igor’s Oberwolfach or Utrecht talks?

• created 2-site with the material from Mike’s web (as he suggested). Added pointers to original articles by Ross Street.

• I started a stub on plethysm.

Does anyone know how this mathematical term originated? I hear someone suggested it to Littlewood. But who? And why? And what’s the etymology, exactly?

• I have a query for Mike, or anyone who wants to tackle it, over at locally finitely presentable category. Mike seems to be saying that only the category of models of a finitary essentially algebraic theory is locally finitely presentable, but some paper seems to suggest otherwise...
• I’m struggling to further develop the page on Schur functors, which Todd and I were building. But so far I’ve only done a tiny bit of polishing. I deleted the discussion Todd and I were having near the top of the page, replacing it by a short warning that the definition of Schur functors given here needs to be checked to see if it matches the standard one. I created a page on linear functor and a page on tensor power, so people could learn what those are. And, I wound up spending a lot of time polishing the page on exterior algebra. I would like to do the same thing for tensor algebra and symmetric algebra, but I got worn out.

In that page, I switched Alt to $\Lambda$ as the default notation for exterior algebra. I hope that’s okay. I think it would be nice to be consistent, and I think $\Lambda$ is most widely used. Some people prefer $\bigwedge$.

• Hello everyone

I am new the nForum and have been informed that my additions to the nLab have introduced terminology clashes and could disrupt the coherence of the nLab. My sincerest apologies to anyone who could be negatively effected. The new pages I introduced follow:

* AbTop
* AbTor
* Alg(T)
* Aut
* Ban
* Beh
* BiComp
* BiTop
* Bij
* BooRng
* BooSpa
* Bor
* CAT
* CAT(X)
* CPO

Also started added pages after reading the nLab page 'database of categories'.
• I have created a new entry center of an abelian category. Maybe it is superfluous as it is just a special case of a construction at center. However in this context there isa number of special theorems which I plan to enter at some point later, so maybe it is not an error to have a separate entry.

• I wrote the beginnings of an article real closed field. I also wrote fundamental theorem of algebra, giving the proof essentially due to Artin which applies generally to real closed fields. Lucky for me, Toby recently wrote quadratic formula! :-)

Things like this have a tendency of spawning a bunch of new articles, but I left out a bunch of potential links in these articles. Please feel free to insert some!

• I’ve redirected the new article stuff to stuff, structure, property, because all of that stuff (pun not originally intended, but kept with delight) is already there, and it didn’t seem like the author knew about it. It doesn’t have to be that way, however, so move stuff > history back to stuff if you disagree, but then make some prominent links between the articles too.

• A $\mathbb{C}-$linear category is simply a category where every Hom(x, y) is a complex vector space and the composition of morphisms is bilinear. A *-category is a $\mathbb{C}-$linear category that has a *-operation on each Hom(x, y) (same axioms a for a *-algebra) and a $C^*-$category further has a norm on each Hom(x, y) that turns it into a Banach space with $s^* s = |s|^2$ and $|st| \leq |s| |t|$ for all arrows s, t (s and t composable).

Is there already a page on the nLab that describes this structure?