nLab
crossed profunctor

Contents

Definition

Given crossed modules 𝕏=(βˆ‚ X:X 1β†’X 0)\mathbb{X}=(\partial_X : X_1\to X_0), 𝕐=(βˆ‚ Y:Y 1β†’Y 0)\mathbb{Y}=(\partial_Y:Y_1\to Y_0) (actions surpressed from the notation), a crossed profunctor

𝕏→𝕐 \mathbb{X}\to\mathbb{Y}

consists of a group PP and morphisms x 1:X 1→Px_1:X_1\to P, x 0:P→X 0x_0:P\to X_0, y 1:Y 1→Py_1:Y_1\to P, y 0:P→Y 0y_0:P\to Y_0

X 1 Y 1 β†˜ x 1 y 1↙ βˆ‚ X↓ P ↓ βˆ‚ Y ↙ x 0 y 0β†˜ 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 }

such that

  • the two triangles commute, x 0∘x 1=βˆ‚ Xx_0\circ x_1=\partial_X, y 0∘y 1=βˆ‚ Yy_0\circ y_1=\partial_Y,

  • the diagonals compose to identites y 0∘x 1=1y_0\circ x_1 = 1, x 0∘y 1=1x_0\circ y_1 = 1,

  • and

    x 1( x 0(p)x)=px 1(x)p βˆ’1,y 1( y 0(p)y)=py 1(y)p βˆ’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∈X 1x\in X_1, p∈Pp\in P, y∈Y 1y\in Y_1.

The complex Y 1→P→X 0Y_1\to P\to X_0 is called the NE-SW complex, and X 1→P→Y 0X_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→P→X 0Y_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 β€œtwisted multiplication” (according to the crossed module page), is there some analogous β€œtwisting” of a profunctor to obtain a crossed profunctor?

Beppe says: no. Actually the name β€œcrossed 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 β€œlarge” 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).
Revised on December 20, 2011 17:47:12 by Beppe Metere (159.149.43.46)