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A monoid in the Category Cat is a strict monoidal category according to Wikipedia. Is it possible to weaken the monoid so that its realisation in Cat is a weak monoidal category? Do we shift up a dimension, and throw in a 2-morphism associator for the monoid satisfying the analogue of Mac Lane's Pentagon, and similarly for the identities?

Or is it still possible to get a weak monoidal category using a monoid by changing the category we realise it in from Cat to some modification of it?

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Yes, you can define a pseudomonoid in any monoidal 2-category, such that a pseudomonoid in the 2-category Cat is precisely a non-strict monoidal category. The definition of monoidal category, interpreted in terms of functors and natural isomorphisms, i.e. in terms of the 2-category Cat, tells you exactly how to define a pseudomonoid.

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