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Uday has added :
Mac Lane, VII.4, only requires a monoidal category to define actions. – at action.
This takes up an old point that I made but never felt up to following up.
Yes, we have actions of group objects there, which require a cartesian monoidal category, since that’s what group objects require. But monoid objects require only a monoidal category, and that’s all that their actions require.
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