Given crossed modules , (actions surpressed from the notation), a crossed profunctor
consists of a group and morphisms , , ,
the two triangles commute, , ,
the diagonals compose to identites , ,
where , , .
The complex is called the NE-SW complex, and is called the NW-SE complex.
If to the above definition added the property that the SE-NW sequence 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.
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).
Revised on December 20, 2011 17:47:12
by Beppe Metere