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. :)

  • $\begingroup$ And I'm not sure if higher-category-theory is an appropriate tag for this question, help retagging would be appreciated. $\endgroup$ – Josh Chen Apr 3 '14 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 Apr 3 '14 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 Chen Apr 3 '14 at 9:58
  • $\begingroup$ Indeed, I mean enriched in abelian groups. $\endgroup$ – Fernando Muro Apr 3 '14 at 10:00
  • 1
    $\begingroup$ 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 Apr 6 '14 at 16:10

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.