The free compact closure of a symmetric monoidal category

Abstract

We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original category. The symmetric monoidal category embeds via a faithful monoidal functor into its completion, but in contrast to the non-symmetric case, this embedding is not full. Our construction factors through the Int construction, which yields another free construction: the free traced monoidal category on a symmetric monoidal category.