i finally understand ⅋ i think. the sense in which it is dual to ⊗ is that when constructing A ⊗ B, A and B must have disjoint origin contexts. when consuming A ⅋ B, you must have disjoint destination contexts, essentially having each branch of the disjunction handled by a separate context.
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i think the reason both are so hard for me to disambiguate is that i am implicitly using the sense of “multiple separate resources”, but separate resources must be joined by something! so either they are in a multiplicative conjunction or a multiplicative disjunction depending on which side of the sequent we are working with.
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its easier to understand the duality of & and ⊕. where ⊕ is a linear version of a classic enum, you can promote A -> A ⊕ B or B -> A ⊕ B, and to destruct it you handle both branches. the opposite of that is to construct A & B you need to create both A and B from the same context, and to destruct it you can choose which A & B -> A or A & B -> B to use.
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coral castle
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