"Tensor product" $\otimes$ of monoids

Question

Referring to the top answer in this post: https://mathoverflow.net/questions/19004/is-the-category-commutative-monoids-cartesian-closed

Would it be reasonable to explain what is a tensor product in this context? I only have an introductory level of experience with category theory and have never come across a tensor product (in any sort of context) until reading this. If this is not something that can be explained in concise way, reference to book I can begin learning from to gain the background I should have for understanding would be appreciated.

Answer 1

The basic notion here is that of a monoidal category, which is a category $C$ together with a functor $\otimes : C \times C \to C$, the monoidal operation, together with some extra data satisfying some extra properties. Any category with finite products is a monoidal category with $\otimes$ the product functor; these are the cartesian monoidal categories.

But there are many important examples which are not cartesian monoidal, the prototypical one being the category $\text{Vect}$ of vector spaces. $\text{Vect}$ can be made into a monoidal category where the monoidal operation is the tensor product of vector spaces, and this is why sometimes monoidal operations are called tensor products. If you aren't familiar with the tensor product of vector spaces, I strongly recommend becoming comfortable with that special case before trying to understand monoidal categories in general.