Completeness and expressiveness for gs-monoidal categories
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Formalised in the study of symmetric monoidal categories, string diagrams are a graphical syntax that has found applications in many areas of Computer Science. Our work aims at systematising and expanding what could be thought of as the core of this visual formalism for dealing with relations and partial functions. To this end, we identify gs-monoidal categories and their graphical representation as a convenient, minimal structure that is useful to formally express such notions. More precisely, to show that such structures naturally arise, we prove that the Kleisli category of a strong commutative monad over a cartesian category is gs-monoidal. Then, we discuss how other categories providing a formalisation of "partial arrows", such as p-categories and restriction categories, are related to gs-monoidal categories. This naturally introduces a pre-order enrichment on gs-monoidal categories, and an equivalence of arrows, called "gs-equivalence": we conclude presenting a completeness result of this equivalence for models defined as lax functors to 𝐑𝐞𝐥.
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