Dear Ellis, On 03/12/2010, at 1:55 AM, Ellis D. Cooper wrote:
(1) Is strict monoidal category the same as monoid in category of categories?
Yes.
(2) Is it not true that in a strict monoidal category if $X\xrightarrow{f}Y\xrightarrow{g}Z$ then $f\square g= g\circ f$?
If I understand correctly, you have arrows f:X->Y and g:Y->Z and you are comparing the tensor products f@g:X@Y->Y@Z and g@f:Y@X->Z@Y. They have different domain and codomain, so cannot be equal. If you considered a commutative monoid in the category of categories, then these arrows would be equal. But such commutative monoids are very rare.
(3) Is the pentagon axiom automatically satisfied in a strict monoidal category?
Yes. In that case it asserts that two identity arrows with the same domain and codomain are equal. Steve Lack.
Many thanks for your patience and pointers.
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