nLab
butterfly

Idea

For $G$ and $H$ two strict 2-groups, their deloopings are strict one-object 2-groupoids $BG$ and $BH$ and a general morphism $f:G\to H$ of 2-groups is by definition a morphism

$$Bf:BG\to BH$$\mathbf{B}f : \mathbf{B}G \to \mathbf{B}H

i.e. a 2-functor – in general a weak one.

A butterfly diagram is a way to describe such weak 2-functors in terms of morphisms between the ordinary groups appearing in the crossed modules corresponding to $G$ and $H$.

Definition

A butterfly or papillon is a crossed profunctor

$$𝕏\to 𝕐$$\mathbb{X} \to \mathbb{Y}

between crossed modules $𝕏=({\partial }_X:X_1\to X_0)$ and $𝕐=({\partial }_Y:Y_1\to Y_0)$ (actions surpressed from the notation), given by a diagram of groups

$$\begin{array}{ccccc}{X}_1& & & & Y_1\\ & {\searrow }^{x_1}& & {}^{y_1}\swarrow \\ {}^{{\partial }_X}\downarrow & & P& & {\downarrow }^{{\partial }_Y}\\ & {\swarrow }_{x_0}& & {}_{y_0}\searrow \\ X_0& & & & Y_0\end{array}$$\array{ X_1 &&&& Y_1 \\ & \searrow^{\mathrlap{x_1}} & & {}^{\mathllap{y_1}}\swarrow \\ {}^{\mathllap{\partial_X}}\downarrow && P && \downarrow^{\mathrlap{\partial_Y}} \\ & \swarrow_{\mathrlap{x_0}} && {}_{\mathllap{y_0}}\searrow \\ X_0 &&&& Y_0 }

satisfying the properties of a crossed profunctor, and in addition such that the NE-SW sequence is exact i.e. a (nonabelian in general) group extension sequence.

Related concepts

Butterflies corresponds to weak functors between the corresponding $2$-groups. A butterfly is flippable, or reversible, if both diagonals are group extensions. There is also a straightforward generalization for $2$-group stacks.

Under the correspondence between crossed modules and categories internal to Grp, butterflies are precisely the saturated anafunctors internal to $\mathrm{Grp}$, using the Grothendieck pretopology of surjective homomorphisms.

References

Butterfly between strict 2-groups have been introduced in

• Behrang Noohi, On weak maps between 2-groups, arXiv

• Behrang Noohi, E. Aldrovandi, Butterflies I: morphisms of 2-group stacks, arXiv, Advances in Mathematics, 221, (2009), 687–773.

• Behrang Noohi, E. Aldrovandi, Butterflies II: Torsors for 2-group stacks,arXiv

Notice that a “torsor over a 2-group stack” is another term for principal 2-bundle (2-truncated principal ∞-bundle) in a (∞,1)-topos of ∞-stacks over some site.