Heriot-Watt Mathematics Report Series
HWM09-3, 13 May 2009
Characterizations of Morita equivalent inverse semigroups
J Funk, M V Lawson and B Steinberg
Abstract
We prove that strong Morita equivalence of inverse semigroups is the same as Morita equivalence
and also to the classifying toposes of the inverse semigroups being equivalent.
In addition, two inverse semigroups are Morita equivalent precisely when they have a joint regular semigroup enlargement
or a joint ordered groupoid enlargement when the inverse semigroups are viewed as inductive groupoids.
Two inverse semigroups are also Morita equivalent when their Cauchy completions are equivalent.
Finally, we show that equivalence bimodules, which witness strong Morita equivalence, can be viewed as abstract atlases.
In this way, we connect with the pioneering work of V.~V.~Wagner on the theory of inverse semigroups and Anders Kock's more recent work on pregroupoids.
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