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I am reading the coherence theorem of monoidal categories. However I am confused by the following paragraph on page 165 of the book "Categories for the working mathmatician" I am reading

$\bf{Here \ are \ my \ questions:}$

(1) He says that the diagrams built up from instances of $\alpha, \lambda, \rho$ by multiplications $\Box$ may not commutes, could anyone provide me an example?

(2) He says that every formal diagram commuts. What is the difference between the formal diagrams and the diagrams built up from instances of $\alpha, \lambda, \rho$ by multiplications $\Box$?

(3)In the sequel, He construced a free monoidal category. I think the process looks very similar to the process of constructing a free monoid by one letter. But he used much space to prove every formal diagram commutes. I want to know what is the significance of free monoidal category? Is it to show every formal diagram commutes?

Thank you for any help.

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  • $\begingroup$ For (1), I think what he means is that there may be monoidal categories in which $A\otimes B= C\otimes D$, but "for silly reasons", where $A,B,C,D$ are parenthesized expressions constructed from certain objects, and thus if you replace $A\otimes B$ by $C\otimes D$ in a diagram and use the $\alpha,\lambda$'s of $C,D$, where it should have been those of $A,B$, then there's no reason that it should commute (because the equality is "silly") $\endgroup$ – Max Jan 19 at 12:38
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(1) There is a famous Isbell's example (see CFWM, p.164): take $\mathbf{Set}_0$, the skeleton of the category of sets, then for every infinite set $X$ the associator $\alpha_{X,X,X}\colon(X\times X)\times X\to X\times(X\times X)$ cannot be equal to the identity $id_X\colon X\to X$.

(2) A formal diagram is a diagram, which is constructed without using the fact that formally different objects in vertices of a diagram are equal. In the previous example, there was the diagram: $$ (X\times X)\times X\xrightarrow{\alpha_{X,X,X}}X\times(X\times X)\xrightarrow{id_X} X, $$
which is constructed using the fact that $X\times(X\times X)=X$, which is an equality between formally different objects.

(3) Free monoidal categories are directly connected with the coherence issues. The coherence theorem for monoidal categories is equivalent to the statement that the free monoidal category generated by an arbitrary set is a preorder. Mac Lane uses another strategy of the proof: he proves the statement for the free monoidal category, generated by the one-elemented set (the set of objects of such free monoidal category is the free magma, generated by the two-elemented set {variable,unit}), and then applies this to the new category $It(B)$, what allows him to prove the whole statement. The proof seems long because it is quite detailed.

I don't know if this is appropriate, but for the proof using free categories generated by the set you can see postings at my categorical weblog (in russian): [1] (construction of the free magma), [2] (for semigroupoidal categories), [3] and [4] (for monoidal categories).

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