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I am reading this introduction to tensor products, which is clearly written:
Whereas in the world of vector spaces, tensors have a clearly visualizable representations, things become more subtle when we generalize to modules over a ring.
There isn’t a simple picture of a tensor (even an elementary tensor) analogous to how a vector is an arrow. Some physical manifestations of tensors are in the previous answer, but they won’t help you understand tensor products of modules. Nobody is comfortable with tensor products at first. Two quotes by Cathy O’Neil and Johan de Jong nicely capture the phenomenon of learning about them:
O’Neil: After a few months, though, I realized something. I hadn’t gotten any better at understanding tensor products, but I was getting used to not understanding them. It was pretty amazing. I no longer felt anguished when tensor products came up; I was instead almost amused by their cunning ways.
de Jong: It is the things you can prove that tell you how to think about tensor products. In other words, you let elementary lemmas and examples shape your intuition of the mathematical object in question. There’s nothing else, no magical intuition will magically appear to help you “understand” it.
This is discouraging. Can we do better than this?
There is the construction of the tensor product as the quotient of enormous (free) module by an enormous sub-module, but it doesn't register with my intuition very well.
Regarding this, Conrad says:
From now on forget the explicit construction of M ⊗R N as the quotient of an enormous free module FR(M × N). It will confuse you more than it’s worth to try to think about M ⊗R N in terms of its construction.
He says instead to use the universal mapping property to understand the tensor product. But I don't like the idea of abandoning the definition of something in order to understand it.
Is this a case where it only makes sense to understand things though its morphisms? I hope not, because I like objects as well as arrows :)