Abstract
We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F: C ⇄ D :G and transformations Id C ⇒GF and Id D ⇒FG , and (2) their exponentials ! M and ! N are related by distributive laws ϱ:! N F⇒F! M and η:! M G⇒G! N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N . We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. nonalternated, backtracking vs. repetitive, uniform vs. nonuniform.
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