Coherence in SMCCS and equivalences on derivations in IMLL with unit

Abstract We study the coherence, that is the equality of canonical natural transformations in non-free symmetric monoidal closed categories ( smcc s). To this aim, we use proof theory for intuitionistic multiplicative linear logic ( imll ) with unit. The study of coherence in non-free smcc s is reduced to the study of equivalences on terms (representing morphisms) in the free category, which include the equivalences induced by the smcc structure. The free category is reformulated as the sequent calculus for imll with unit so that only equivalences on derivations in this system are to be considered. We establish that any equivalence induced by the equality of canonical natural transformations over a model can be axiomatized by some set of “critical” pairs of derivations. From this, we derive certain sufficient conditions for full coherence, and establish that the system of identities defining smcc s is not Post-complete: extending this system with an identity that does not hold in the free smcc does not in general cause the free smcc to collapse into a preorder. In order to give a larger context to these results, we study the equality of canonical morphisms in non-free symmetric monoidal categories, and establish that w.r.t. a broad subclass of smcc s, the equivalences induced by the equality of canonical natural transformations over a model coincide with the equivalences induced by the equality of canonical morphisms for all interpretations in that model.

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