Calculi of string diagrams are increasingly used to present the syntax andalgebraic structure of various families of circuits, including signal flowgraphs, electrical circuits and quantum processes. In many such approaches, thesemantic interpretation for diagrams is given in terms of relations orcorelations (generalised equivalence relations) of some kind. In this paper weshow how semantic categories of both relations and corelations can becharacterised as colimits of simpler categories. This modular perspective isimportant as it simplifies the task of giving a complete axiomatisation forsemantic equivalence of string diagrams. Moreover, our general result unifiesvarious theorems that are independently found in literature and are relevantfor program semantics, quantum computation and control theory.
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