nLab
crossed profunctor

Definition

Given crossed modules $𝕏=({\partial }_X:X_1\to X_0)$, $𝕐=({\partial }_Y:Y_1\to Y_0)$ (actions surpressed from the notation), a crossed profunctor

$$𝕏\to 𝕐$$\mathbb{X}\to\mathbb{Y}

consists of a group $P$ and morphisms $x_1:X_1\to P$, $x_0:P\to X_0$, $y_1:Y_1\to P$, $y_0:P\to Y_0$

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

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

  $$x_1({}^{x_0(p)}x)=p{x}_1(x)p^{-1},y_1({}^{y_0(p)}y)=p{y}_1(y)p^{-1},$$x_1 ({}^{x_0(p)}x) = p x_1(x) p^{-1}, \,\,\,\,\,y_1 ({}^{y_0(p)}y) = p y_1(y)p^{-1} \,,

  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.

Related concepts

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 “twisted multiplication” (according to the crossed module page), is there some analogous “twisting” of a profunctor to obtain a crossed profunctor?

Beppe says: no. Actually the name “crossed profunctor” 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.

References

This notion appeared in

• M. Jibladze, Coefficients for cohomology of “large” categories , pp. 169–179, in H. Inassaridze (Ed.), K-theory and Homological Algebra (A Seminar held at the Razmadze Mathematical Institute in Tbilisi, Georgia, USSR 1987-88), Lec. Notes in Math. 1437, Springer 1990 (jibladzeCoeffLargeCats.djvu).