Proof Theory for Full Intuitionistic Linear Logic , Bilinear Logic , and Mix Categories

This note applies techniques we have developed to study coherence in monoidal categories with two tensors, corresponding to the tensor–par fragment of 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 noncommutative logic; we also consider the noncommutative version of FILL. The essential difference between FILL and BILL lies in requiring that a certain tensorial strength be an isomorphism. In any FILL category, it is possible to isolate a full subcategory of objects (the “nucleus”) for which this transformation is an isomorphism. In addition, we define 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 allow non-logical axioms. We define 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 introduced the notion of “weakly distributive category”, now renamed “linearly distributive category”, in order to study the pure proof theory of the cut rule for the sequent calculus with finite 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 of as the tensor–par fragment of linear logic with no negation. We wished to show how 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 Research partially supported by NSERC, Canada. Research partially supported by Le Fonds FCAR, Québec, and NSERC, Canada. Diagrams in this paper were produced with the help of the TEXcad drawing program of G. Horn and the diagram macros of F. Borceux. Received by the editors 1996 September 23 and, in revised form, 1997 March 17. Published on 1997 March 28 1991 Mathematics Subject Classification : 03B70, 03F07, 03G30, 18D10, 18D15, 19D23.