Inverse Sets and Inverse Correspondences Over Inverse Semigroups

In this paper, we introduce notions called inverse set and inverse correspondence over inverse semigroups. These are analogies of Hilbert C^*-modules and in the C^*-algebra theory. We show that inverse semigroups and inverse correspondences form a bicategory. In this bicategory, two inverse semigroups are equivalent if and only if they are Morita equivalent.