Characterisations of Morita equivalent inverse semigroups

For a fixed inverse semigroup S, there are two natural categories of left actions of S: the category Fact of unitary actions of S on sets X meaning actions where SX = X, and the category Étale of étale actions meaning those unitary actions equipped with a function p : X → E(S), to the set of idempotents of S, such that p(x)x = x and p(sx) = ses∗, where s∗ denotes the inverse of s. The category Étale can be regarded as the classifying topos of S. There is a forgetful functor U from Étale to Fact that forgets étale structure and simply remembers the action. Associated with these two types of actions are appropriate notions of Morita equivalence which we term Morita equivalence and strong Morita equivalence, respectively. We prove three main results: first, strong Morita equivalence is the same as Morita equivalence; second, the forgetful functor U has a right adjoint R, and the category of Eilenberg-Moore algebras of the monad M = RU is equivalent to the category of presheaves on the Cauchy completion C(S) of S; third, we show that equivalence bimodules, which witness strong Morita equivalence, can be viewed as abstract atlases, thus connecting with the pioneering work of V. V. Wagner on the theory of inverse semigroups and Anders Kock’s more recent work on pregroupoids. 2000 Mathematics Subject Classification: 20M18, 18B25, 18B40.