During lecture, my abstract algebra professor said that the exactness of the tensor product is "absolutely essential" to the existence of physical phenomena such as black holes and the big bang.
Is it more or less directly related to the existence of such phenomena or is it a big stretch to make such a conclusion?
If the former, what is the connection?
The essence of tensor is that the product doesn't change under coordinate transformations. If the product changes, the corresponding physical system may lack certain consistence.
It is quite philosophical that some phenomena exits only if you can properly describe it, independent of the angle you look at it, or the language you use.