nLab butterfly

Idea

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

$$\mathbf{B}f : \mathbf{B}G \to \mathbf{B}H$$

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 $H$.

There is a more general version describing weak morphisms among internal crossed modules in semi-abelian categories.

Definition

A butterfly or papillon is a crossed profunctor

$$\mathbb{X} \to \mathbb{Y}$$

between crossed modules $\mathbb{X}=(\partial_X : X_1\to X_0)$ and $\mathbb{Y}=(\partial_Y:Y_1\to Y_0)$ (actions suppressed from the notation), given by a diagram of groups

$$\array{ X_1 &&&& Y_1 \\ & \searrow^{\mathrlap{x_1}} & & {}^{\mathllap{y_1}}\swarrow \\ {}^{\mathllap{\partial_X}}\downarrow && P && \downarrow^{\mathrlap{\partial_Y}} \\ & \swarrow_{\mathrlap{x_0}} && {}_{\mathllap{y_0}}\searrow \\ X_0 &&&& Y_0 }$$

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 $Grp$, using the Grothendieck pretopology of surjective homomorphisms.

In fact under the same correspondence, butterflies correspond to locally representable profunctors, while split butterflies give representable profunctors. This extends to the intrinsic setting, where the base category is semi-abelian (see references below).

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,Advances in Mathematics, 225, (2010), 922-976.

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.

• Omar Abbad, Sandra Mantovani, Giuseppe Metere, Enrico M. Vitale, Butterflies in a semi-Abelian context, arxiv/1104.4275