We introduce the notion of a crossed module over an inverse semigroup which generalizes the notion of a module over an inverse semigroup in the sense of Lausch [1], as well as the notion of a crossed module over a group in the sense of Whitehead [4] and Maclane [3]. With any crossed \( S \)-module \( A \) we associate a \( 4 \)-term exact sequence of inverse semigroups \( A \xrightarrow{i} N \xrightarrow{\beta} S \xrightarrow{\pi} T \), which we call a crossed module extension of \( A \) by \( T \). We then introduce the so-called admissible crossed module extensions and 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 \( H^3_\le(T^1,A^1) \), whenever \( T \) is an \( F \)-inverse monoid.

This is a joint work [2] with Mikhailo Dokuchaev and Mayumi Makuta (both from the University of São Paulo).

[1] Lausch, H. Cohomology of inverse semigroups. J. Algebra 35 (1975), 273 - 303.
[2] Dokuchaev, M., Khrypchenko, M., and Makuta, M. Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups. J. Algebra 593 (2022), 341 - 397.
[3] MacLane, S. Cohomology theory in abstract groups. III. Operator homomorphisms of kernels. Ann. Math. (2) 50 (1949), 736 - 761.
[4] Whitehead, J. H. C. Combinatorial homotopy. II. Bull. Am. Math. Soc. 55 (1949), 453 - 496