Models of Cyclic Sharing Theory

Abstract

We have delayed introducing the models for cyclic sharing graphs until now. The main reason is that, while the models of acyclic settings are obtained by revising well known concepts from category theory, we need a relatively new notion for interpreting cyclic bindings — traced monoidal categories introduced by Joyal, Street and Verity [50]. The notion of trace, while the concept itself goes back to the classical traces of linear maps between finite dimensional vector spaces, has originally been invented for analyzing cyclic structures arising from mathematics and physics, notably the interplay of low-dimensional topology (knot theory) and quantum groups (e.g. [78, 51]); it is then a natural idea to use this concept for modeling our cyclic graph structure too, and it does work.