I am reading Lang's Algebra, especially Tensor product, which he defines as follows:
Let $R$ be a ring, and $M_1,\ldots,M_n,F$ be $R$-modules. A map $f:M_1\times \cdots \times M_n\rightarrow F$ is multilinear if it is $R$-linear in each variable/component $M_i$. One may view the multilinear maps of a fixed set of modules $M_1,\ldots,M_n$ as the objects of a category: if $f:M_1\times \cdots \times M_n\rightarrow F$ and $g:M_1\times \cdots \times M_n\rightarrow G$ are multilinear, we define a morphism $f\rightarrow g$ to be a homomorphism $h:F\rightarrow G$ such that $g=h\circ f$ (first apply $f$, then $h$). A universal object in this category is called a Tensor product of $M_1,\ldots,M_n$.
My Question: In what sense should one think tensor product as universal? More precisely, is it universally attracting or repelling?
Why question came to mind?: After mentioning about tensor product as a universal object, I went to see the definition he gave. He defined universally attracting and repelling objects in a Category, and said,
When the context makes our meaning clear, we shall call universally attracting or repelling objects as universal.
Therefore, I asked myself:
for the category of multilinear maps from a fixed set of $n$ modules over $R$, how should we take the universal object? If only universally attracting object exists in this category, why universally repelling does not exist (or vice versa)?
Best Answer
In fact, the tensor product is universally repelling (or, as S.Farr comments, an initial object). Your category also has a universally attracting (or terminal) object. This is the map $0 : M_1\times \cdots \times M_n\rightarrow 0$ to the trivial $R$-module $0$. As you see, this is not very interesting.
It also shows that initial and terminal objects, if they exist in a category, are not necessarily isomorphic. See also Universally repelling and universally attracting objects are equivalent?