Re: Severe Strict Monoidal Category Naivete

[Steve Lack <steve.lack@mq.edu.au>]

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.
> 
> 


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]