How can one understands the definition of tensor from the purely formal point of view?
To what abstract structure this concept can be generalized?
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It depends upon what your base generality is. Monoidal categories are a vast generalization: http://en.wikipedia.org/wiki/Monoidal_category – Alex Youcis Sep 17 '13 at 09:52
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Possible duplicate of http://math.stackexchange.com/questions/10282/an-introduction-to-tensors – isomorphismes Sep 17 '13 at 18:03