Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups (Q2062737)

The authors introduce the notion of a \textit{crossed module} over an inverse semigroup. With any crossed \(S\)-module \(A\), they associate a \(4\)-term exact sequence of inverse semigroups \(A \xrightarrow{i} N \xrightarrow{\beta} S \xrightarrow{\pi} T\), which is called a \textit{crossed module extension of \(A\) by \(T\)}. It is shown that any such extension induces a \(T\)-module structure on \(A\), and equivalent extensions induce the same \(T\)-module structure on \(A\). It is then constructed a map from the set \(\mathcal{E}(T,A)\) of equivalence classes of crossed module extensions of a \(T\)-module \(A\) by \(T\) to the (Lausch) inverse semigroup cohomology group \(H^3(T^1,A^1)\). Under this map, the set \(\mathcal{E}_\le(T,A)\) of equivalence classes of the so-called \textit{admissible} crossed module extensions of \(A\) by \(T\) is mapped to the group of order-preserving cohomology \(H^3_\le(T^1,A^1)\). For the converse map, with any cocycle from \(Z^3_\le(T^1,A^1)\) the authors associate a crossed module extension \(A \xrightarrow{i} N \xrightarrow{\beta} S \xrightarrow{\pi} T\) of \(A\) by \(T\), which gives rise to a map from \(H^3_\le(T^1,A^1)\) to \(\mathcal{E}(T,A)\). Here \(S\) is the \textit{\(E\)-unitary cover} of \(T\) through the free group \(FG(T)\) and \(N\) is a semilattice of groups which can be seen as a direct product of \(A\) and \(K=\pi^{-1}(E(T))\) in the appropriate category. The main result of the paper states that \(H^3_\le(T^1,A^1)\) is mapped bijectively onto \(\mathcal{E}_\le(T,A)\), whenever \(T\) is an \textit{\(F\)-inverse} monoid.