I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$. (This is defined as the category with objects all formal finite direct sums $\bigoplus_i x_i$ of objects $x_i$ in $\mathcal{C}$, and morphisms $$f\colon \bigoplus_{i=1}^n x_i \rightarrow \bigoplus_{j=1}^m y_j$$ being $m\times n$ matrices with columns indexed by $x_i$ and rows by $y_j$, where the $(j,i)$-th entry is a morphism $f_{ij}\colon x_i \rightarrow y_j$. Composition is given by matrix multiplication.)

What conditions (if any) must $\mathcal{C}$ satisfy in order for its tensor product to lift to a tensor product in $\mathcal{M}at\,\mathcal{C}$?

Since we simply formally added the direct sums I almost feel like saying that in the case that $\mathcal{C}$ is monoidal we should just add the requirement that $\otimes$ of objects distributes over $\oplus$, then taking $\otimes$ of morphisms to be the Kronecker product of matrices (with composition in place of multiplication) seems to make everything work out.

Is it alright for me to construct a tensor product for $\mathcal{M}at\,\mathcal{C}$ from the tensor product of $\mathcal{C}$ in this way?

And is there (or is this) a general construction that makes the additive envelope of a monoidal category itself monoidal?

I have this page off the nLab to work off, but other than that have not much else of a clue where to look, so some introductory references would also be highly appreciated. :)

1more comments