Given crossed modules $\mathbb{X}=(\partial_X : X_1\to X_0)$, $\mathbb{Y}=(\partial_Y:Y_1\to Y_0)$ (actions surpressed from the notation), a crossed profunctor
consists of a group $P$ and morphisms $x_1:X_1\to P$, $x_0:P\to X_0$, $y_1:Y_1\to P$, $y_0:P\to Y_0$
such that
the two triangles commute, $x_0\circ x_1=\partial_X$, $y_0\circ y_1=\partial_Y$,
the diagonals compose to identites $y_0\circ x_1 = 1$, $x_0\circ y_1 = 1$,
and
where $x\in X_1$, $p\in P$, $y\in Y_1$.
The complex $Y_1\to P\to X_0$ is called the NE-SW complex, and $X_1\to P\to Y_0$ is called the NW-SE complex.
If to the above definition added the property that the SE-NW sequence $Y_1\to P\to X_0$ is exact in that it is a (nonabelian in general) group extension, this is called a butterfly. See there for more details.
Alex asks: Is there any relation between a profunctor and a Crossed Profunctor? Just as one could look at a crossed module as a module with โtwisted multiplicationโ (according to the crossed module page), is there some analogous โtwistingโ of a profunctor to obtain a crossed profunctor?
Beppe says: no. Actually the name โcrossed profunctorโ was not a happy choice. A crossed profunctor is just the normalized version of a profunctor, as a crossed module is the normalization of a groupoid. This extends to a strong biequivalence between the bicategory of crossed modules, crossed profunctors and their morphisms (in Grp) and that of crossed modules, profunctors and their morphisms.
This result holds also in intrinsic settings, say when we consider internal categories in a semi-abelian category.
This notion appeared in