> Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) A tensor isn'

> Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-) A tensor isn't actually a multidimensional array, just as a linear map isn't actually a matrix. Defining tensors as the multidimensional abstraction of scalars, vectors and matrices to n indices is anti-pedagogical, though regrettably common because it's computationally useful. Learning tensors this way is like learning linear algebra primarily by doing matrix multiplication. It hollows out all of what's actually going on by focusing on notation and symbol pushing. What you're comparing here is the heavily symbol-laden, notation-encumbered definition of a tensor with the elegant abstraction of a momoid. If you'd like to compare them on the merits of their abstraction, an alternative definition of "tensor" would be better. The object is not its representation. For example: a tensor is a multilinear map between a product of vector spaces and another vector space. So T: V_{1} x V_{2} x ... x V_{n} --> W is a tensor. Furthermore, every tensor T is itself an element of a tensor product. An example of a tensor product is R^{nk}, which is the nk-dimensional real space you obtain by taking the tensor direct product of the n- and k- dimensional real spaces, R^{n} ⊗ R^{k}. In this way the tensor product is an abstraction of the familiar Cartesian product in which the dimension of two spaces is multiplied instead of the cardinality of two sets. This definition makes it easier to understand why the objects of a monoidal category can be considered tensors. A monoidal category is a category C where the functor ⊗ is defined such that C x C --> C. Note the similarity of this definition with my alternative definition of a tensor - no multidimensional arrays are required! You're just moving up one level of generality from linear maps and Cartesian products. Technically speaking, the monoidal category functor is actually an abstraction the tensor product, the way the tensor product is an abstraction of the Cartesian product. Whereas many monoidal categories admit a Cartesian product, some monoidal categories (like the category of vector spaces) admit a tensor product. Edit - Sorry for the bracket notation used in subscripts and superscripte, I was thinking in LaTeX...