Chain complexes of symmetric categorical groups

We define the cohomology categorical groups of a complex of symmetric categorical groups, and we construct a long 2-exact sequence from an extension of complexes. As special cases, we obtain Ulbrich cohomology of Picard categories and the Hattori–Villamayor–Zelinsky sequences associated with a ring homomorphism. Applications to simplicial cohomology with coefficients in a symmetric categorical group, and to derivations of categorical groups are also discussed.