We give a definition of categorical model for the multiplicative fragment of non-commutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting.
It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the Tannaka-Krein reconstruction theorem is extended to the entropic setting.
Keywords: Linear logic, monoidal categories, Hopf algebras.
2000 MSC: 03F07, 03F52, 18A15, 18D10, 57T05.
Theory and Applications of Categories, Vol. 10, 2002, No. 17, pp 424-460.