Braided Premonoidal Coherence

通用伪态射及其在辫子与对称幺半函子的图解融贯性中的应用

Universal pseudomorphisms, with applications to diagrammatic coherence for braided and symmetric monoidal functors

本文提出通用伪态射理论，证明在一定假设下伪态射融贯性等价于严格代数融贯性，并应用于普通、对称与辫子幺半函子的图解融贯性。

Coherence for braided and symmetric pseudomonoids

Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in the theory of monoids and commutative monoids, and generalise standard coherence theorems for braided and symmetric monoidal categories.

Loop spaces, and coherence for monoidal and braided monoidal bicategories

We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.

Natural Associativity without the Pentagon condition

A premonoidal category is equipped only with a bifunctor and a natural isomorphism for associativity. We introduce a (deformation) natural automorphism representing the deviation from the Pentagon condition. We uncover a binary tree representation for all diagrams involving associativity natural isomorphisms and (deformation) natural automorphisms and provide a link to permutations and linear orderings. This leads to other notions of premonoidalness. We define these notions and prove coherence results for each.

An alternative description of braided monoidal categories

We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to be equivalent to the usual associativity, unit and braiding axioms. We also discuss the next dimensional version, that is, b-structures on bicategories. As an application, we show how special b-categories result in the Yang-Baxter equation, and how special b-bicategories produce Zamolodchikov's tetrahedron equation. Finally, we define a cohomology theory (the…

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 was answered by Laplaza whe…