Braided distributivity

TL;DR

This paper develops new coherence conditions for distributivity in categories with one symmetric and one braided monoidal structure, crucial for modeling anyon systems in topological quantum computing.

Contribution

It introduces and proves the validity of novel coherence conditions for distributivity when one monoidal structure is braided, extending Laplaza's symmetric case.

Findings

Proposed coherence conditions are both necessary and sufficient.

Conditions hold in categories modeling anyons with additive structures.

Identified a redundancy in Laplaza's original conditions.

Abstract

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 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 conditions are (a) strong enough to imply Laplaza's 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 found a new redundancy in Laplaza's conditions.