Let be a semigroup. A left -act is called ‘closed’ if the map given by is surjective and injective. The full subcategory (of all left -acts and all -homomorphisms) of all closed left -acts is denoted by -. Two semigroups and are called ‘Morita equivalent’ if the categories - and - are equivalent. A semigroup is called an ‘enlargement’ of its subsemigroup if and . A ‘consolidation’ of a strongly connected category is a map , , such that . A small category equipped with a consolidation can be made into a semigroup by defining where has domain and has codomain .
It is proved that semigroups with local units and are Morita equivalent if and only if one of the following conditions is satisfied: 1) the Cauchy completions and are equivalent; 2) and have a joint enlargement which can be chosen to be regular if and are both regular; 3) there is a unitary Morita context with surjective mappings; 4) there is a consolidation on and a local isomorphism .
Semigroups with local units Morita equivalent to a semigroup satisfying certain additional conditions (for example, to be a group, inverse semigroup, semilattice or orthodox semigroup) are described as well.