Coherence for braided distributivity.

In category-theoretic models for the anyon systems proposed for topological quantum computing, the essential ingredients are two monoidal structures, $oplus$ and $otimes$. The former is symmetric but the latter is only braided, and $otimes$ is required to distribute over $oplus$. What are the appropriate coherence conditions for the distributivity isomorphisms? We came to this question working on a simplification of the category-theoretical foundation of topological quantum computing, which is the intended application of the research reported here. This question above was answered by Laplaza when both monoidal structures are symmetric, but topological quantum computation depends crucially on $otimes$ being only braided, not symmetric. We propose coherence conditions for distributivity in this situation, and we prove that our coherence conditions are (a) strong enough to imply Laplazau0027s when the latter are suitably formulated, and (b) weak enough to hold when --- as in the categories used to model anyons --- the additive structure is that of an abelian category and the braided $otimes$ is additive. Working on these results, we also found a new redundancy in Laplazau0027s conditions.