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June 23, 2008

Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

Posted by Urs Schreiber

guest post by Timothy Porter


At the meeting in Barcelona, on Thursday morning, we had Behrang Noohi talking on “Butterflies and morphisms between weak 2-groups”. This was a fun expanded version of his preprint, (see arXiv:math/0506313).

The idea is that as 2-groups and categorical groups are the same, what do lax monoidal functors between 2-groups look like? This is important if we want to handle equivalences between 2-groups in a constructive and efficient way.

His viewpoint is that often crossed modules are a neat and effective way of viewing 2-groups as they tend to be fairly small, so he looked for the analogues of lax morphisms between crossed modules in order to obtain the groupoid Hom wk(G,H)\Hom_{wk}(\mathbf{G},\mathbf{H}) of such weak maps in simple terms.

The first translation of this was very ‘cocycly’ and difficult to digest, (that was his point in giving it), but then after the coffee break (and some excellent little croissantty things!) he introduced us to ‘butterflies’. (I doubt that I can manage to produce adequate diagrams for the blog so will try to explain butterflies without them.)

Suppose G=(G 1G 0)\mathbf{G} = (G_1\to G_0) and H=(H 1H 0)\mathbf{H} = (H_1\to H_0) are two crossed modules, then a butterfly from G\mathbf{G} to H\mathbf{H} is a diagram with the two crossed modules down the two vertical sides, another group in the middle and two diagonal sequences (NW-SE and NE-SW) (i)G 1κEρH 0(i)\quad G_1\stackrel{\kappa}{\to}E\stackrel{\rho}{\to}H_0 and (ii)H 1ιEσG 0.(ii) \quad H_1\stackrel{\iota}{\to}E\stackrel{\sigma}{\to}G_0.(Draw it correctly and you will see why it is a butterfly!)

Both sequences are complexes (i.e. the composite maps are trivial) and (ii) is exact. There are some compatibility conditions to be satisfied as well, but I will leave those out.

Initially I could not see what this definition was doing, but Behrang gave an excellent trip through the theory and I now think this is an very important direction to pursue.

I like to think things simplicially so for me crossed modules are also simplicial groups with Moore complex of length one, and the homotopy category of crossed modules is obtained inverting weak equivalences of such things. It therefore involves spans GweEH\mathbf{G}\stackrel{we}{\leftarrow}\mathbf{E}\to \mathbf{H} There is a link here with the butterflies, since one result that Behrang mentioned is that a butterfly provides one with a natural choice of E\mathbf{E}, namely a certain crossed module (G 1×H 1E)(G_1\times H_1\to E). This is sort of a minimal choice.

He mentioned extensions to weak maps between 2-crossed modules and also some applications to Deligne’s theorem on Picard stacks, and another to `gerbes bound by a crossed module’. This latter thread was handled by Ettore Aldrovandi in the afternoon. Bruce says he will write something about that so I will not do so here. I found that connection of very great interest as it derives from work by someone called Debremaeker in the 1970s and I spent some time exploring this earlier this year. But for that it is ‘over to Bruce’.

Posted at June 23, 2008 1:17 PM UTC

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Read the post Aldrovandi on Non-Abelian Gerbes and 2-Bundles
Weblog: The n-Category Café
Excerpt: A talk by Aldrovandi on different perspectives on gerbes and 2-bundles.
Tracked: June 23, 2008 9:09 PM

Re: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

One neat feature of butterflies is that the homotopy cokernel of a butterfly can be constructed using the sequence (i) that Tim has above.

Posted by: David Roberts on June 24, 2008 1:54 AM | Permalink | Reply to this

Re: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

I’m very happy to read summaries of some talks that I foolishly skipped in order to write summaries of the other talks.

My silence of late is caused by limited access to the internet here in Granada. Yesterday I visited the Alhambra and took lots of photographs of tilings, which will eventually appear here.

But besides the Alhambra, Granada is also famous for its experts on categorical groups, or 2-groups! Pilar Carrasco, Antonio Cegarra, Antonio Garzón, Manuel Bullejos and Aurora del Río are all here, and I’ve learned some interesting things. In particular, Carrasco says she plans to translate her thesis on hypercrossed complexes into English, at the request of Jim Stasheff. This is very good news, since these structures give a lot of insight into homotopy types, and the published paper on them is rather terse.

Posted by: John Baez on June 26, 2008 10:30 AM | Permalink | Reply to this

Re: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

That is very good news.

Posted by: Tim Porter on June 26, 2008 1:42 PM | Permalink | Reply to this

Re: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

Both the title of blog entry and the title of Behrang Noohi’s talk on the conference webpage suggest his talk was about ‘morphisms between weak 2-groups’. However, Tim Porter’s description of the talk, and the paper it’s based on, lead me to believe that Noohi’s talk was actually about ‘weak morphisms between strict 2-groups’ — where the strict 2-groups are described as crossed modules.

So, it seems some sort of evil typo is propagating itself through cyberspace. If so, it must be stopped!

The 2-category of strict 2-groups, weak morphisms and weak 2-morphisms is equivalent to the really interesting 2-category, which consists of weak 2-groups, weak morphisms and weak 2-morphisms. So, it’s worth getting a handle on, and that’s what Noohi did.

Posted by: John Baez on July 12, 2008 3:50 PM | Permalink | Reply to this

Re: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

John is, of course, right. `Propagate’ is also the right term here. I copied and pasted the title from the conference web page!

He is also completely right about the importance of understanding the weak morphisms between strict 2-groups.

Posted by: Tim Porter on July 13, 2008 4:40 PM | Permalink | Reply to this

Re: Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups

More on butterflies:

Butterflies I: morphisms of 2-group stacks, by Ettore Aldrovandi and Behrang Noohi.

Posted by: David Corfield on August 28, 2008 12:56 PM | Permalink | Reply to this

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