Tensor versus Kronecker on Matrices

Two “tensor products” live on matrices, and they are not the same map. The functorial monoidal tensor is defined only up to a chosen identification of index sets; the Kronecker product is a concrete blocking of entries. This note records how they differ, the comparison map between them, and exactly when the coordinate passage between the categories and respects either one.

Tensor versus Kronecker

The functorial tensor on sends to and to ; on it is defined only after choosing a bijection of index sets. The Kronecker product instead blocks entries directly, with component rule

Here rows of are indexed by and columns by , so a factor pair is baked into the layout.

The comparison map

There is a canonical comparison matching the abstract tensor to the Kronecker blocking. It behaves differently on and off the diagonal.

Proposition on isomorphism only along the same-space diagonal

is a natural isomorphism precisely on the diagonal , where the dimensions are equal. Off the diagonal it stays injective on morphisms but fails to be injective on objects. An object of dimension has as many Kronecker factorisations as has ordered factor pairs , so cannot reconstruct the factors and cannot be inverted there.

The coordinate embedding

Construction of the forgetful functor and its section

Fixing bases gives a forgetful , , and a section , . The section is full and faithful and injective on objects, witnessed by , since a linear map between coordinate spaces is exactly its matrix.

When the embedding respects the product

Because , coordinates turn the abstract tensor into the concrete blocking. Thus is strictly monoidal for and only monoidal up to for . The general shape of the obstruction is a preservation lemma.

Lemma on when a functor preserves a monoidal product

A functor preserves the monoidal structure iff there exist correcting endofunctors (adjusting the two arguments, the product, and the result) with naturally. For the coordinate embedding the correction is exactly . With the collapse to identities, whereas with they do not.