A monoid in the the Category Cat is a strict monoidal category according to wikipedia. Is it possible to weaken the monoid so 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 Maclanes Pentagon, and similar for the identities? 

Or is 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.