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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. :)

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  • $\begingroup$ And I'm not sure if higher-category-theory is an appropriate tag for this question, help retagging would be appreciated. $\endgroup$
    – Josh
    Commented Apr 3, 2014 at 7:54
  • $\begingroup$ I've corrected that tag. I guess you should add that $\mathcal C$ is additive. Your construction looks correct to me, but I don't know of any reference. $\endgroup$
    – Fernando Muro
    Commented Apr 3, 2014 at 9:41
  • $\begingroup$ Thanks @Muro! :) I don't think $\mathcal{C}$ is required to be additive, I thought the point of taking the additive envelope was to construct an additive category containing the original. But I think you're right in that I need $\mathcal{C}$ to be enriched over abelian groups. Thanks! $\endgroup$
    – Josh
    Commented Apr 3, 2014 at 9:58
  • $\begingroup$ Indeed, I mean enriched in abelian groups. $\endgroup$
    – Fernando Muro
    Commented Apr 3, 2014 at 10:00
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    $\begingroup$ just a note: you say that you consider direct sums, i.e. coproducts, but your morphisms are really morphisms from the coproduct of x_i's to the product of y_i's, and then to compose you identify them, so really they are "formal biproducts" $\endgroup$
    – Adam Gal
    Commented Apr 6, 2014 at 16:10

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Yes, your construction will work. Additive envelope is a functor $$\mathbf{Env} : \mathbf{Ab\text{-}Cat} \to \mathbf{Add},$$ from the category of Abelian enriched categories to the category of additive categories. This functor has a monoidal structure given by the obvious functors $$\mathbf{Env}(A)\otimes \mathbf{Env}(B) \to \mathbf{Env}(A\otimes B),$$ which on objects acts as $$\Big(\bigoplus_ix_i, \bigoplus_iy_i\Big) \mapsto \bigoplus_{i,j}(x_i, y_j).$$

Thus, $\mathbf{Env}$ take monoids in $\mathbf{Ab\text{-}Cat}$, which are abilian enriched monoidal categories, to monoids in $\mathbf{Add}$, i.e. monoidal additive categories. It also follows that this construction is functorial.

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