Morita等价逆半群的刻画
Characterizations of Morita equivalent inverse semigroups
摘要：
证明了分别由$C^{\ast}$ -代数理论、拓扑理论、半群理论和序群胚理论所激发的逆半群的Morita等价的四种不同概念是等价的。我们还证明了逆半群的酉作用范畴在étale作用范畴上是一元的。因此，逆半群的酉作用范畴等价于其柯西完备化上的预层范畴。更一般地，我们证明了同样的情况下，闭行动的范畴，这是用来定义的森田理论在半群理论，任何半群的右局部单位。
英文摘要：
We prove that four different notions of Morita equivalence for inverse semigroups motivated by, respectively, $C^{\ast}$-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.
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