The n-Category Café

Hmm, would Urs’ construction of the universal $G$-bundle in terms of the nerve of $\mathrm{INN}(G)$

I should point out that the insight that the realization of the nerve of the groupoid which I know by the various names $$\mathrm{INN}(G) = G // G = \mathrm{Codisc}(G) = (G \stackrel{\mathrm{Id}}{\to} G)$$ is a model for $E G$ in that $$|\cdot| : (G \to G // G \to \Sigma G) \mapsto (G \to E G \to BG)$$ is due to Segal. I originally learned this from Danny Stevenson, who also pointed me to the relevant article:

G. B. Segal, Classifying spaces and spectral sequences, Publ. Math. IHES No. 34 (1968) 105-112 .

But I like to make the point that one can regard the groupoid $G // G$ as “playing the role of” the universal $G$-bundle even without passing to geometric realizations, but stayin entirely within the world of groupoids. And that this point of view simplifies and illuminates some points.

So, for instance, the fact that $G // G$ “is” the universal $G$-bundle is, in the world of groupoids, encoded by the facts that

a) $G // G$ it is (not equal but) equivalent to the trivial groupoid. This yields the fact that $E G$ is (not equal but) homotopy equivalent to the point, i.e. contractible.

ii) $G // G$ has in fact the structure of a 2-group (this is an important piece of information which becomes pretty much invisible when you pass to $E G$ in the usual way, but by Segal’s construction, $E G$ has in fact the structure of a topological group!, in contrast to $B G$, which is a group only if $G$ is abelian). Using this and the canonical injection $$G \to G // G$$ we get the $G$-action on $E G$. And the fact that $G \to G // G \to \Sigma G$ is exact then gives that this action is free.

In the very last section of my article with David Roberts we indicate more ways to that that and how $G // G$ plays the role of the universal $G$-bundle, without the need of ever passing to geometric realizations.

And using this point of view, we then find the analogous statements for 2-groups. David and I never ever try to geometrically realize this stuff here. But John Baez and Danny Stevenson have a paper almost finished (compare the remarks John made here) where they do consider the universal $G_{(2)}$-bundle realized as a topological space.

shed any light on the Thom Space $M G$?

I wish I knew the answer to that. I am lacking intuition for Thom spaces. Maybe if there is a “nice” way to think of them, one could try to make an educated guess for whether and how there is a way to think of them just on the level of groupoids. That would be cool.

John said:

(In England such people are called swans: they look so elegant as they float upstream, but down beneath their little feet are busily paddling away.)

Well wangling theory is one of my favorites, but I don’t know if I have deep insight into it (there’s also the danger of cramps when moving feet too rapidly underwater).

In this case there was an announcement some time of ago of a yearlong program at IAS, which I heard of through the grapevine (sound effect of busy paddling), and it was eventually on the IAS website, at which point one could apply for a semester or year of membership (with a deadline sadly about a semester ago). Once you hear about it it’s a regular application procedure like a postdoc or something, no mystery there (maybe there’s a mystery in the selection process, but my sense was there weren’t so so many applicants specifically for the program – a “benefit” of not advertising too broadly!).

I imagine that there are still openings for shorter term visits - I would recommend writing directly to Roman Bezrukavnikov, who’s running the program, or to Mary Jane Hayes, who (as School Administrative Officer) runs everything. Of course the weeklong conferences are wide open and advertising now for visitors.. but for longer stays I think contacting an organizer is the way to go.

Maybe I should clarify that when I talk about openings, this has to do with institute housing (and salary). They are quite happy (within bounds of course) to have people come and attend things and interact, the most limited resource is the Institute housing. In ‘97 I simply tagged along with my advisor to the QFT and Math program, even got an office and building key, just had to find my own apartment in town (and I think some students will be doing so next year). So if the housing is not an issue I think this should be quite feasible.

I do find that one always hears about such things far too late (eg I’ve never seen a conference poster before registration actually closes) — that’s one reason to have adverts in blogs like this, and to frequent some places that list conferences – I like Ravi Vakil’s list in algebraic geometry for example (and theoretically the Notices listing, though that’s so broad I never have the patience to scan through for things I might like). It might be nice for people following this blog if there was a feature with a list of upcoming conferences in related areas that people could contribute to?