# Definition of a *Monoidal Abelian Category*

Does the definition of a monoidal abelian category require any coherence between the abelian structure and the monoidal structure?

• In this paper, 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 $\textrm{Hom}(A, C) \otimes_{\mathbb{Z}} \textrm{Hom}(B, D) \to \textrm{Hom}(A \otimes B, C \otimes D)$. – Zhen Lin Jul 25 '12 at 15:27
• Sorry but I don't see why you're tensoring over ${Z}$. – MikhailMatrix Jul 25 '12 at 15:40
• Because $\textrm{Hom}(X, Y)$ is an abelian group if we have an abelian category. – Zhen Lin Jul 25 '12 at 16:37
• Obviously 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. – Omar Antolín-Camarena Aug 1 '12 at 18:40