# Tensor product as universal object: a question arising from Lang's Algebra

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.

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)?

• The tensor product is meant to be the initial object in this category. Commented Aug 10, 2021 at 7:48
• Why we should not consider terminal object in such category? Commented Aug 10, 2021 at 7:49
• Initial and terminal aren't really inherently different. Just take the opposite category. The tensor product is sort of frustrating categorically since it is not in generally a product or coproduct. Commented Aug 10, 2021 at 7:53
• The category you describe is the so called 'under category' of the product of the $M_i$ and R-modules, and as such inherits all limits from R-Mod (see ncatlab.org/nlab/show/under+category,). Thus the terminal object - as it is a limit - would not be that interesting. Commented Aug 10, 2021 at 7:55
• I never studied category theory, but from the standpoint of linear algebra, a tensor generalises every object one could encounter in this field: scalar, vector, linear map, linear form, inner product, etc. So, calling a tensor a universal object still makes a lot of sense to me. Commented Aug 10, 2021 at 7:55

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.