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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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  • $\begingroup$ What are pomonoids? $\endgroup$
    – David White
    Mar 29, 2021 at 12:48
  • $\begingroup$ @DavidWhite Monoids with a left/right compatible partial order $\endgroup$
    – misseuler
    Mar 29, 2021 at 13:20
  • $\begingroup$ @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. $\endgroup$
    – Benjamin Steinberg
    Mar 29, 2021 at 14:24

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