# Commutative monoidal category free over singleton? Useful in proving coherence?

The proof of coherence in monoidal categories in CWM is based on the existence of a monoidal category free over a singleton. Denoting this category by $\mathcal{W}=\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho}\right)$ it can be observed that $\mathcal{W}_{0}$ is a thin groupoid. Its objects are so-called 'binary words'. For every pair $u,v\in\mathcal{W}_{0}$ the homset $\mathcal{W}_{0}\left(u\square v,v\square u\right)$ contains exactly one arrow, and denoting it with $\hat{\gamma}_{u,v}$ it seems to me that $\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho},\hat{\gamma}\right)$ can be recognized as a commutative monoidal category. My questions are:

1) Can $\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho},\hat{\gamma}\right)$ be classified as a commutative monoidal category free over a singleton?

2) If the answer on the first question is 'yes' then can coherence in commutative monoidal categories be proved the same way (used in CWM) as in the proof for monoidal categories?

I think that I am overlooking complications, because the proof in CWM of coherence for monoidal categories appears to be more complex.

• The coherence theorem for monoidal categories is, in some sense, an explicit description of the free monoidal category generated by any category, not just the singleton. – Zhen Lin Mar 31 '14 at 13:05
• @ZhenLin If I am not mistaking then the coherence proof for monoidal categories is in essence the theorem stating that 'for any monoidal category $\mathcal B$ and any object $b\in\mathcal B$, there is a unique morphism $\mathcal W\rightarrow \mathcal B$ of monoidal categories with $-\rightarrow b$'. Here '$\{-\}$' is the singleton. In CWM I encountered monoidal categories free over a set. Not free over a category. My vision is beyond doubt smaller than yours, though. – drhab Mar 31 '14 at 13:56
• Free on a singleton is too small to yield anything useful. Free on a set of objects is better, but in practice what one is really using is free on a category. – Zhen Lin Mar 31 '14 at 13:58
• @ZhenLin Implicitly you are saying that the theorem mentioned in my former comment (CWM page 162) is not in essence the coherence proof for monoidal categories. Is that correct? – drhab Mar 31 '14 at 14:15
• Having it for the singleton is very far from enough. That would be like concluding that every monoid is commutative, because the free monoid on one generator is... – Zhen Lin Mar 31 '14 at 14:34