Conclusion On The Abstract Approach
In principle the abstract approach can recover everything done via the traditional approach. In practice this may not seem so simple. On the other hand the notion of natural is consistent with the general covariance principle of general relativity. The latter deals with tensor fields (tensors varying from point to point on a manifold), but covariance asserts that the language of tensors is essential to the proper formulation of general relativity.
Some decades later the rather abstract view coming from category theory was tied up with the approach that had been developed in the 1930s by Hermann Weyl (in his book The Classical Groups). In a way this took the theory full circle, connecting once more the content of old and new viewpoints.
Read more about this topic: Multilinear Algebra
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