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• 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?

• the entry fibrations of quasi-categories was getting too long for my taste. I have to change my original plans about it.

Now I split off left Kan fibration from it, which currently duplicates material from this entry and from fibration fibered in groupoids. I'll see how to eventualy harmonize this a bit better.

Presently my next immediate goal is to write out as a pedagogical introduction to the notion of left/right fibration a nice detailed proof for the fact that a functor is an op-fibration fibered in groupoids precisely if its nerve is a left Kan fibration.

I wanted to do that today, but got distracted. Now I am getting too tired. So I'll maybe postpone this until tomorrow...

• I added material to Young diagram, which forced me to create entries for special linear group and special unitary group. I also added a slight clarification to unitary group.

I would love it if someone who knows algebraic geometry would fix this remark at general linear group:

Given a commutative field $k$, the general linear group $GL(n,k)$ (or $GL_n(k)$) is the group of invertible $n\times n$ matrices with entries in $k$. It can be considered as a subvariety of the affine space $M_{n\times n}(k)$ of square matrices of size $n$ carved out by the equations saying that the determinant of a matrix is zero.

In fact it’s ’carved out’ by the inequality saying the determinant is not zero… so its description as an algebraic variety is somewhat different than suggested above. Right???

• Started on bibundles, but there seem to be a raft of competing definitions. Perhaps they're all special cases of a most general definition.

• I did a little bit of rewriting and cleaning up at reflective subcategory, in an effort to make things clearer for the neophyte. Part of the cleaning-up was to remove a query initiated by Zoran under the section Characterizations (I rewrote a bit to make the question vanish altogether).

There’s another query of Zoran at the bottom which I think was answered by Mike, but let me ask before removing it.

• Aleks Kleyn emailed me saying he would like a reference or two to work on Ω-groups, so maybe someone can help him out. I put a query on the relevant page.
• started TCFT

eventually we should better reflect at the relevant nLab entries that Costello’s classification result of 2d TCFTs – proving the corresponding conjecture by Kontsevich – was the precurser of the proof of the full cobordism hypothesis: Hopkins and Lurie tried to generalize Costello’s proof (they had not actually heard of Baez-Dolan back then).

• Just got the following query from Harald Hanche-Olsen about the page separation axioms. As I’ve never seen that notation before either (but agree with Harald’s comments in both parts), I’m forwarding it here so that the person who first adopted it (Toby?) or others can chip in.

I hadn’t seen the notation $\stackrel\circ\ni$ for a neighbourhood before, but it looks like a reasonable notation that I might want to adapt. BUT it seems more appropriate for a neighbourhood of a point rather than a neighbourhood of a set. Wouldn’t $\stackrel\circ\supset$ or $\stackrel\circ\supseteq$ be more appropriate for that case? What is the rationale for the usage on that page?

• edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.

And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?

I think the entry-situation here deserves to be further harmonized.