Is a weak monoidal category a monoid object is some category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:54:50Z http://mathoverflow.net/feeds/question/26317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26317/is-a-weak-monoidal-category-a-monoid-object-is-some-category Is a weak monoidal category a monoid object is some category? Shamim 2010-05-28T22:31:57Z 2010-05-28T23:15:41Z <p>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? </p> <p>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.</p> http://mathoverflow.net/questions/26317/is-a-weak-monoidal-category-a-monoid-object-is-some-category/26322#26322 Answer by Mike Shulman for Is a weak monoidal category a monoid object is some category? Mike Shulman 2010-05-28T23:15:41Z 2010-05-28T23:15:41Z <p>Yes, you can define a <em>pseudomonoid</em> 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.</p>