Initial objects in the category of bilinear maps $M \times N \to L$. where $M,N$ are fixed $R$-modules, and $L$ arbitrary $R$-module.

In our class we are looking at the definition of tensor product, and here’s a (paraphased) remark that I don’t get.

Let $M,N$ be fixed $R$-modules, and $L$ an arbitrary $R$-module. At this point we only know that the tensor product $M \bigotimes_R N$ is a $R$-module, such that there exists a bilinear map $\Phi: M \times N \to M \bigotimes_R N$.

Let $\mathscr{C}_{M,N}$ be the category of bilinear maps $M \times N \to L$.

Then $\text{Hom}_R(M \bigotimes_R N, L)$ (the set of $R$-linear maps M $\bigotimes_R N \to L$) being isomorphic to $\text{Hom}_R(M, N; L)$ (the set of bilinear maps $ M \times N \to L$) is equivalent to $\Phi$ being an initial object in $\mathscr{C}_{M,N}$.

I interpret this as saying that, given an object $g$ in $\mathscr{C}_{M,N}$, there exists only one morphism $\Phi \to g$. How does $\text{Hom}_R(M \bigotimes_R N, L) \simeq \text{Hom}_R(M,N; L)$ imply this?

Solution 1:

In what follows, $R$ is a commutative ring with unity.

Given two $R$-modules $M$ and $N$, a tensor product of $M$ and $N$ is an $R$-module $M \otimes_R N$ together with a bilinear map $\otimes \colon M \times N \to M \otimes_R N$ that has the following property:

For any $R$-module $L$, and any bilinear map $b \colon M \times N \to L$, there exists a unique linear map $l \colon M \otimes_R N \to L$ such that $l \circ \otimes = b$.

Observe that if $X$ and $Y$ are sets, a function $f \colon X \to Y$ is bijective if and only if for any $y \in Y$ there exists a unique $x \in X$ such that $f(x)=y$. With this in mind, observe that the above property is equivalent to the following:

For any $R$-module $L$, the function $$\begin{align*} \operatorname{Hom}_R(M \otimes_R N,L) & \longrightarrow \operatorname{Bil}_R(M,N;L) \\ l & \longmapsto l \circ \otimes \end{align*}$$ is bijective. (Indeed, it is an isomorphism of $R$-modules, but this doesn't matter here).

Finally, define the category $\mathscr B_{M,N}$ whose objects are bilinear maps $M \times N \to *$ (the codomain is any $R$-module), and for any two objects $b_1 \colon M \times N \to L_1$ and $b_2 \colon M \times N \to L_2$, a morphism $l \colon b_1 \to b_2$ is the same as a linear map $l \colon L_1 \to L_2$ such that $l \circ b_1 = b_2$. Thus, the defining property of the tensor product is also equivalent to the following:

For any object $b$ of $\mathscr B_{M,N}$ there exists a unique morphism $l \colon \otimes \to b$. In other words, $\otimes$ is an initial object of $\mathscr B_{M,N}$.