I came across this MO post and it got me thinking. In monoidal categories we can define module and bi-module categories, so what can we say about morita equivalence in this situation? Which of the Morita theorems still hold, and do the equivalent statements still hold for categories without a ring-like structure such as semigroups or pomonoids?
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What are pomonoids? – David White Mar 29 '21 at 12:48
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@DavidWhite Monoids with a left/right compatible partial order – misseuler Mar 29 '21 at 13:20
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@DavidWhite, a pomonoid is a monoid with a stable partial order. Morita theory for monoids was studied by Knauer and Banachevski and behaves very well. It is of course a special case of the results for categories. For semigroups Telwar developed a theory and he and others eventually got a satisfactory theory for semigroups with $S=SS$. This is also where people can do things for nonunital rings. Valdis Laan and others have developed the theory for pomonoids but I am not an expert. – Benjamin Steinberg Mar 29 '21 at 14:24