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I think this is an important question, and the grammatical errors in the question's formulation---while easily tolerated by someone well-versed in English---could be misleading. I therefore did some mainly grammatical copy-editing. No distortions of meaning or even style of the question.
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edit approved Jul 11, 2017 at 20:39
Peter Heinig
edit approved Jul 11, 2017 at 20:39
Peter Heinig
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Is a weak monoidal category a monoid object isin some category?  

A monoid in the the Category Cat is a strict monoidal category according to wikipediaWikipedia. 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 MaclanesMac Lane's Pentagon, and similarsimilarly 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.?

Is a weak monoidal category a monoid object is some category?  

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.

Is a weak monoidal category a monoid object in some category?

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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Shamim
Shamim
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Is a weak monoidal category a monoid object is some category?

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.