In this paper we introduce and study the categorical group of derivations, Der(G, A), from a categorical group G into a braided categorical group (A,c) equipped with a given coherent left action of G. Categorical groups provide a 2-dimensional vision of groups and so this object is a sort of 0-cohomology at a higher level for categorical groups. We show that the functor Der(-, A) is corepresentable by the semidirect product of A with G and that Der(G,-) preserves homotopy kernels. Well-known cohomology groups, and exact sequences relating these groups, in several different contexts are then obtained as examples of this general theory.
Keywords: derivation, categorical group, cohomology
2000 MSC: 18D10, 18G50, 20J05, 20L05
Theory and Applications of Categories,
Vol. 13, 2004,
No. 5, pp 86-105.
http://www.tac.mta.ca/tac/volumes/13/5/13-05.dvi
http://www.tac.mta.ca/tac/volumes/13/5/13-05.ps
http://www.tac.mta.ca/tac/volumes/13/5/13-05.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/13/5/13-05.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/13/5/13-05.ps