The n-Category Café

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 $\Hom_{wk}(\mathbf{G},\mathbf{H})$ 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 $\mathbf{G} = (G_1\to G_0)$ and $\mathbf{H} = (H_1\to H_0)$ are two crossed modules, then a butterfly from $\mathbf{G}$ to $\mathbf{H}$ 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) $$(i)\quad G_1\stackrel{\kappa}{\to}E\stackrel{\rho}{\to}H_0$$ and $$(ii) \quad H_1\stackrel{\iota}{\to}E\stackrel{\sigma}{\to}G_0.$$(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 $$\mathbf{G}\stackrel{we}{\leftarrow}\mathbf{E}\to \mathbf{H}$$ 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 $\mathbf{E}$, namely a certain crossed module $(G_1\times H_1\to E)$. 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’.