Coherence of the Double Negation in Linear Logic

Many formulations of proof nets and sequent calculi for Classical Linear Logic (CLL) [7, 8] take it for granted that a type A is identical to its double negation A⊥⊥. On the other hand, since Seely [13], it has been assumed that ∗-autonomous categories [1, 2] are the appropriate semantic models of (the multiplicative fragment of) CLL. However, in general, in a ∗-autonomous category an object A is only canonically isomorphic to its double involution A∗∗. For instance, in the category of finite dimensional vector spaces and linear maps, a vector space V is only isomorphic to its double dual V ∗∗. This raises the questions whether ∗-autonomous categories do not, after all, provide an accurate semantic model for these proof nets and whether there could be semantically non-identical proofs (or morphisms), which must be identified in any system which assumes a type is identical to its double negation. Whether this can happen is not completely obvious even when one examines purely syntactic descriptions of proofs with the isomorphism between A and A⊥⊥ present such as [11, 9] or the alternative proof net systems of [4] which are faithful to the categorical semantics. Fortunately, there is no such semantic gap: in this talk we provide a coherence theorem on the double involution on ∗-autonomous categories, which tells us that there is no difference between the up-to-identity approach and the up-to-isomorphism approach, as far as this double-negation problem is concerned. Theorem. Any free ∗-autonomous category is strictly equivalent to a free ∗-autonomous category in which the double-involution (−)∗∗ is the identity functor and the canonical isomorphism A ' A∗∗ is an identity arrow for all A.