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Coherence of proof-net categories. (English) Zbl 1274.03089

Summary: The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly and Mac Lane’s coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence theorem is also proved in the presence of arrows corresponding to the mix principle of linear logic. The notion of proof-net category catches the unit-free fragment of the notion of star-autonomous category, a special kind of symmetric monoidal closed category.

MSC:

03F07 Structure of proofs
03F52 Proof-theoretic aspects of linear logic and other substructural logics
18A10 Graphs, diagram schemes, precategories
18A15 Foundations, relations to logic and deductive systems
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
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