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

Bf:BGBH \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 GG and HH.

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


A butterfly or papillon is a crossed profunctor

𝕏𝕐 \mathbb{X} \to \mathbb{Y}

between crossed modules 𝕏=( X:X 1X 0)\mathbb{X}=(\partial_X : X_1\to X_0) and 𝕐=( Y:Y 1Y 0)\mathbb{Y}=(\partial_Y:Y_1\to Y_0) (actions surpressed from the notation), given by a diagram of groups

X 1 Y 1 x 1 y 1 X P Y x 0 y 0 X 0 Y 0 \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.

Butterflies corresponds to weak functors between the corresponding 22-groups. A butterfly is flippable, or reversible, if both diagonals are group extensions. There is also a straightforward generalization for 22-group stacks.

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

In fact under the same correpondence, 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).


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

Revised on December 20, 2011 18:21:40 by Beppe Metere (