Motivation to introduce tensor products

Question

I have to expose what the tensor product of module is, so I'm wondering for a motivation to introduce the definitions and all the respective theorems. For example, (remark that i'm just starting with the concept) Is there any example of how can I describe a set (better if it is a group) of bilinear forms, meaning using the characterization in R-homomorphism, maybe I can say this group of bilinar forms is isomorphic to an specific group... or something like that

Answer 1

There are a few motivations for the tensor product:

• The "free vector space" functor $\mathsf{Set}\to\mathsf{Vect}$ is monoidal with respect to disjoint union $\bigsqcup$ and direct sums $\bigoplus$. It is also monoidal with respect to $\prod$ and ... tensor products $\bigotimes$.
• In quantum theory, a vector space can have a distinguished basis of "pure states," in which case linear combinations of them are superpositions of pure states. To obtain the vector space associated to a composite physical system, we need a basis of pure states which are pairs of pure states of the original two systems.
• Say we want to glue to algebras together, like how $k[x]$ and $k[y]$ may be "glued" in order to obtain $k[x,y]$. This is achieved with tensoring: $k[x]\otimes_k k[y]\cong k[x,y]$.
• We may extend scalars similarly by gluing a "larger" ring to a module, as in the $R$-module $M$ becomes the $S$-module $S\otimes_R M$, where $R$ is a subring of $S$.
• Its universal property: the space of bilinear maps $A\times B\to C$ is naturally isomorphic to the hom space $\hom(A\otimes B,C)$. This generalizes to multilinear maps.
• The tensor-hom adjunction $\hom(A,\hom(B,C))\cong \hom(A\otimes B,C)$, which is actually a linearized version of "currying" (from the category $\mathsf{Set}$ to the category $\mathsf{Vect}$); in fact the free vector space functor turns the currying identity into the tensor-hom adjunction. A related fact is $\hom(A,B)\cong A^{\ast}\otimes B$.

All of the bullet points above may be generalized in one way or another. For instance, there is also a free functor from the category of $G$-sets to the category of representations of $G$ over a field $k$, which is just the free vector space functor $\mathsf{Set}\to\mathsf{Vect}$ when $G$ is trivial.

You seem to be talking about the universal property in your question.