Understanding the tensor product

I know the definition of the tensor product, and I can somehow understand its importance, but among several constructions in mathematics, somehow I just never grasped the meaning of the tensor product. I don't understand them in the same sense as I understand the concepts of say products, coproducts, semidirect product, fibre product, pushout, kernel, cokernel and their universal properties.

When proving things I have a feeling for when I should take for instance the fibre product, and I have a feeling for when it is appropriate to consider certain structures, but the tensor product just never came naturally to me.

Just to make it clear. I understand the algebra behind the construction of the tensor product. I can operate with it. I can also verify and prove the Hom-Tensor adjunction, although I do not fully grasp what it really is telling me. I also have seen that the right-exactness of tensoring provides useful, but I never really understood what is going on.

Someone told me that I should look at tensor products as linearizing, but I am not sure in what context this was intended as a visualization.

I am wondering if anyone knows a reference or could explain to me what the tensor product really means. I am seeking examples where it is "obvious" that tensoring will help solve a problem, and how it solves the problem.

This is probably a very wide question, and I have not specified the context in which the tensor product is considered, but I feel like there must be some general explanation for what the tensor product really does.

• what about all the differential geometry that this construction catch? Commented Nov 12, 2015 at 23:25
• Think about a multilinear map on a product of vector spaces to $W$ and the universal property of the tensor product. The universal property guarantees a linear map from the tensor product space to $W$. See it as a change to a space where the multilinear map is a linear one (making the space as big as it is needed to get ride of that "multi")
– mdot
Commented Nov 13, 2015 at 0:56
• A very similar question was asked just this morning. You should search the site for other questions on the motivation for the tensor product. Here are two relevant threads: one, two. Commented Nov 13, 2015 at 2:51
• I did search the site, but of course I searched using key words like "intuition" and "understanding", and not "importance", so I was only led to tons of articles asking about details in the construction which is not what I was wondering about. However, I am satisfied with the answers in the first link you sent. Commented Nov 13, 2015 at 6:18
• You just get used to it over some time. When you feel comfortable adding numbers in school and suddenly multiplication of numbers comes in as a new mysterious thing, later you will see that there is nothing mysterious about that. You just have to keep calculating. The same goes for modules, their direct sums and tensor products. Commented Feb 23, 2020 at 15:28