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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.

• finally added to crossed complex…. the definition! :-)

Also added a paragraph on what the crossed complex associated to a strict globular $\infty$-groupoid is.

• 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?