Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups

Abstract We define and study the notion of a crossed module over an inverse semigroup and the corresponding 4-term exact sequences, called crossed module extensions. For a crossed module A over an F-inverse monoid T, we show that equivalence classes of admissible crossed module extensions of A by T are in a one-to-one correspondence with the elements of the cohomology group H ≤ 3 ( T 1 , A 1 ) .