On low dimensional cohomology of crossed modules with nontrivial coefficients

G. Donadze, T. Pirashvili

For a crossed module G* and abelian crossed module A* equipped with an action of G* we construct a truncated cochain complex C*(G*, A*). We show that the corresponding cohomology groups Hi(G*, A*), 0≤i≤2 have several nice properties and can be considered as a "cocycle" description of cotriple cohomology of crossed modules first introduced in [4], and further studied in [9], [15]. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 16(4) (2023), pp. 143-173