Let’s back up then.

A monoidal category consists of

A category $\mathbf{M}$,

Functors $\otimes: \mathbf{M} \times \mathbf{M} \to \mathbf{M}$ and $I: 1 \to \mathbf{M}$ (it may be simpler to think of $I$ as simply an object of $\mathbf{M}$),

Natural isomorphisms $\alpha: \otimes \circ (\otimes \times 1_{\mathbf{M}}) \to \otimes \circ (1_{\mathbf{M}} \times \otimes)$ and $\lambda: \otimes \circ \langle I !, 1_{\mathbf{M}} \rangle \to 1_{\mathbf{M}}$ and $\rho: \otimes \circ \langle 1_{\mathbf{M}}, I ! \rangle \to 1_{\mathbf{M}}$,

satisfying some equations called coherence axioms or coherence conditions.

Thus, the associativity isomorphism $\alpha$ is a natural transformation between functors of the form $\mathbf{M} \times \mathbf{M} \times \mathbf{M} \to \mathbf{M}$. The domain of $\alpha$ is the composite of two functors

$\mathbf{M} \times \mathbf{M} \times \mathbf{M} \stackrel{\otimes \times 1_{\mathbf{M}}}{\to} \mathbf{M} \times \mathbf{M} \stackrel{\otimes}{\to} \mathbf{M}$and the codomain is the composite of two functors

$\mathbf{M} \times \mathbf{M} \times \mathbf{M} \stackrel{1_{\mathbf{M}} \times \otimes}{\to} \mathbf{M} \times \mathbf{M} \stackrel{\otimes}{\to} \mathbf{M}.$The component of $\alpha$ at $(A, B, C) \in Ob(\mathbf{M} \times \mathbf{M} \times \mathbf{M})$, denoted $\alpha_{A, B, C}$, is therefore a morphism of the form $(A \otimes B) \otimes C \to A \otimes (B \otimes C)$. Hopefully that clears up some of the confusion.

In case the notation in the description I wrote for $\lambda$ and $\rho$ is confusing, I’ll spell it out in more detail. Let ’$1$’ denote the terminal category, with exactly one object and exactly one morphism, the identity morphism. Since $1$ is terminal, there is (for a given category $\mathbf{M}$) a unique functor $\mathbf{M} \to 1$; I denote this unique functor as $!: \mathbf{M} \to 1$ because ’!’ is a symbol often used to mean ’unique’. Next, recall that if $\mathbf{B}, \mathbf{C}$ are categories, then there is a product category together with projection functors $\pi_1: \mathbf{B} \times \mathbf{C} \to \mathbf{B}$, $\pi_2: \mathbf{B} \times \mathbf{C} \to \mathbf{C}$, with the universal property that given a category $\mathbf{A}$ and a pair of functors $(F: \mathbf{A} \to \mathbf{B}, G: \mathbf{A} \to \mathbf{C})$, there exists a unique functor $H: \mathbf{A} \to \mathbf{B} \times \mathbf{C}$ such that $\pi_1 \circ H = F$ and $\pi_2 \circ H = G$. This functor $H$ is denoted $\langle F, G \rangle: \mathbf{A} \to \mathbf{B} \times \mathbf{C}$.

So, for example, we can let $F: \mathbf{M} \to \mathbf{M}$ be the composite $\mathbf{M} \stackrel{!}{\to} 1 \stackrel{I}{\to} \mathbf{M}$, and let $G$ be the identity functor $1_\mathbf{M}: \mathbf{M} \to \mathbf{M}$, and then form the functor $\langle I \circ !, 1_\mathbf{M} \rangle: \mathbf{M} \to \mathbf{M} \times \mathbf{M}$. The domain of the natural transformation $\lambda$ is the functor $\mathbf{M} \to \mathbf{M}$ formed as the composite

$\mathbf{M} \stackrel{\langle I \circ !, 1_\mathbf{M} \rangle}{\to} \mathbf{M} \times \mathbf{M} \stackrel{\otimes}{\to} \mathbf{M}$and the codomain is the identity functor $1_\mathbf{M}: \mathbf{M} \to \mathbf{M}$. Then, if you follow this through, the component of $\lambda$ at an object $A \in Ob(\mathbf{M})$ will have the form $I \otimes A \to A$. It is denoted $\lambda_A: I \otimes A \to A$.

The spelling-out of $\rho$ is similar.

Tensors as used in relativity theory, or in differential geometry, can be given a formal mathematical expression in terms of multilinear functions involving modules and their duals over a ring of smooth functions (or more exactly, over a sheaf of rings of smooth functions on a manifold). In some ways the physics might be good training, but I think it’s conceptually much simpler to begin the study of tensors in the context of multilinear functions over a commutative ring, say starting with the ring of integers or the ring of real numbers. If you have Lang’s Algebra book, that might be a good place to start learning about multilinear functions (I’d have to check to be sure); most decent graduate algebra textbooks, or advanced texts on linear algebra, should have something on multilinear algebra.

]]>Yes, I quickly saw the nLab article. I see the same problems as in Wikipedia.

You say “A construct like $A \otimes (B \otimes C)$ is an object of a monoidal category, not a morphism.”

I am still confused: “Two functors $F$ and $G$ are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from $F$ to $G$.” (Wikipedia). But $A \otimes (B \otimes C)$ and $(A \otimes B) \otimes C$ are objects not functors, how can they be naturally isomorphic?

I am going to take study of monoidal categories seriously, because my $\mathsf{pFCD}$ is a tensor product (at least for join-semilattice) and I want to consider its properties induced by the fact that this is a tensor product. I will also try to prove that it is a tensor product for arbitrary posets. Thus I need to read something “serious” about monoidal categories. Thank you for this book, I am going to look into it.

“For infinitary tensor products: here it is important to get clear on what exactly you want or expect.” I suppose that one (or more) of my concepts may be described as infinite tensor products:

- prestaroids
- staroids
- completary staroids

But which of them?

I studied a year of a physics course in a university. There we studied the Einstein’s definition of tensors.

]]>Well, have you read anything relevant in the nLab?

There are a bunch of articles surrounding the concept of monoidal category, although I think that particular article may be overdue for some improvement. The associativity and unit constraints are natural isomorphisms in all of their components, which may address your 4th question.

For students who are serious, I might instead recommend Tom Leinster’s book. In particular, he discusses two ways of presenting $n$-ary tensor products, in terms of either ’biased’ or ’unbiased’ definitions, which might help address your 2nd question.

For infinitary tensor products: here it is important to get clear on what exactly you want or expect. For example, in the category of commutative rings, finite tensor products are actually finite coproducts in that category, and so it would be reasonable to interpret an infinitary tensor product as an infinite coproduct, and one can form such a thing as a filtered colimit of finite tensor products. But for other purposes, this type of infinitary tensor product might not be exactly what one wants. See for example this MO question by Martin Brandenburg. It generally pays to pay attention to the exact formulation of the universal property one is after.

I just went now to the Wikipedia Talk page, and it seems you are confused, especially where you “misunderstand something”. A construct like $A \otimes (B \otimes C)$ is an object of a monoidal category, not a morphism.

I should think it would help you immensely to know some of the basic examples of tensor products from Algebra. In particular, it would help to be comfortable with tensor products of modules over a commutative ring, or tensor products of bimodules over a general ring. Smash products of pointed spaces are another good example.

]]>I have asked several question at this Wikipedia talk page (end of the page):

- Infinite tensor product?
- $n$-ary tensor product for finite $n$?
- I misunderstand something
- Natural isomorphism in which component?

The questions are about Wikipedia but may apply to nLab as well.

Please help to answer all my four questions.

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