Does the definition of a monoidal abelian category require any coherence between the abelian structure and the monoidal structure?
Definition of a *Monoidal Abelian Category*
8
$\begingroup$
category-theory
-
1In [this paper](http://arxiv.org/abs/math/0004160), the author defines an abelian monoidal category to be an abelian category if the functor $(A, B) \mapsto A \otimes B$ is additive in each variable separately. Note that this implies there is a group homomorphism $\textrm{Hom}(A, C) \otimes_{\mathbb{Z}} \textrm{Hom}(B, D) \to \textrm{Hom}(A \otimes B, C \otimes D)$. – 2012-07-25
-
0Sorry but I don't see why you're tensoring over ${Z}$. – 2012-07-25
-
2Because $\textrm{Hom}(X, Y)$ is an abelian group if we have an abelian category. – 2012-07-25
-
1Obviously the answer is that depends one what you're using the notion of *monoidal abelian category* for. But I'd think that the condition Zhen Lin mentions is a very reasonable one. – 2012-08-01