Is the tensor product of a monoidal category associative?

Question

A strict monoidal category is a monoid in $\operatorname{Cat}$, i.e. a category $\mathcal{M}$, together with a functor $\otimes\colon\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ and a distiguished object $U$ such that $\otimes\circ (1\times \otimes)=\otimes\circ (\otimes\times 1)$ and for every object $C\otimes U=C=U\otimes C$.

A monoidal category is a category $\mathcal{M}$, together with a functor $\otimes\colon\mathcal{M}\times\mathcal{M}\to\mathcal{M}$, a distiguished object $U$ and three natural isomorphisms making a pentagon and a triangle commute.

My question is: is the functor $\otimes$ of a monoidal category associative, as it is in a strict monoidal category?

Answer 1

One of the three isomorphisms you refer to is the associator, which makes $\otimes$ associative "up-to-isomorphism." The most important difference between a monoidal and a strict monoidal category is that the associativity is relaxed in this way.

Answer 2

Spelling out each components that makes a strict monoidal category a monoid in (Cat, $\times$, 1) :

To define a monoid in this monoidal category, you need $\mathcal{M}\in Cat$, $\otimes\colon\mathcal{M}\times\mathcal{M}\to\mathcal{M}$, $U: 1\to\mathcal{M}$ and commutation with 3 diagrams inherited from the monoidal structure of (Cat, $\times$, 1).

If you do not identify $U: 1\to\mathcal{M}$ with $U\in \mathcal{M}$, you conditions read out as $\otimes.(U\times id_M) : 1\times M \to M\times M \to M = \lambda_{Cat} : 1\times M \to M $ (and likewise for the association axiom which you did not write)

Whereas the corresponding diagram (in Cat) defining a monoidal category has some (iso) transformation between those 2 functors.

So what you would need is to look as pseudo-monoids, to have exactly the definition of a monoidal category.

If you see $U: 1\to\mathcal{M}$ as $U\in \mathcal{M}$, this means that $\otimes$ is not associative, but that every time you have a way of going from one way of parenthesizing to another, there is a corresponding transformation between the two resulting composed objects.

This idea is captured in the notion of a bicategory, and a indeed (the delooping of) a monoidal category (or a pseudo-monoid in (Cat,$\times$,1)) is just a bicategory with a single object.

You might also stop thinking about 0, 1, 2, 3, etc.. objects to consider how those structures allow you to pick any numbers $n$ of objects, and relate the way they compose : they are all equal or at least related depending if you are in strict, or in normal monoidal categories.