Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories.

This note applies techniques we h a ve developed to study coherence in monoidal categories with two tensors, corresponding to the tensor{par fragment o f linear logic, to several new situations, including Hyland and de Paiva's Full Intuitionistic Linear Logic (FILL), and Lambek's Bilinear Logic (BILL). Note that the latter is a non-commutative logicc we also consider the noncommutative v ersion of FILL. The essential diierence between FILL and BILL lies in requiring that a certain tensorial strength be an isomorphism. I n a n y FILL category, it is possible to isolate a full subcategory of objects (the \nucleus") for which this transformation is an isomorphism. In addition, we deene and study the appropriate categorical structure underlying the MIX rule. For all these structures, we do not restrict consideration to the \pure" logic as we a l l o w non-logical axioms. We deene the appropriate notion of proof nets for these logics, and use them to describe coherence results for the corresponding categorical structures. 0. Introduction In CS91] we i n troduced the notion of \weakly distributive category", now renamed \lin-early distributive category", in order to study the pure proof theory of the cut rule for the sequent calculus with nite sequences of formulas on both sides of the turnstile. This is generally thought of as the \classical" sequent calculus, but in fact this proof theory is not truly \classical" in any real sense, and may be thought o f a s t h e tensor{par fragment of linear logic with no negation. We w i s h e d t o s h o w h o w features could be added in a modular fashion to this basic categorical setting, in order to model the more expressive fragments of linear logic: this program is now largely complete, see CS91, BCST, BCS92], and includes the subject matter of this paper. Crucial to this program was the provision of an intrinsic characterization of the par. In classical linear logic the negation was an obstruction, for it allowed the par to be viewed as merely the de Morgan dual of the usual tensor product, and so for its special