Linearly distributive categories (LDC), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivity. A seminal result in monoidal category theory is the Fox theorem, which characterizes cartesian categories as symmetric monoidal categories whose objects are equipped with canonical comonoid structures. The aim of this work is to extend the Fox theorem to LDCs and characterize the subclass of cartesian linearly distributive categories (CLDC). To do so, we introduce medial linearly distributive categories (MLDC), medial linear functors, and medial linear transformations. The former are LDCs which respect the logical medial rule, appearing frequently in deep inference, and are the appropriate categorical structure at the intersection of LDCs and duoidal categories.
Keywords: linearly distributive categories, cartesian, Fox theorem, medial rule
2020 MSC: 18M45
Theory and Applications of Categories, Vol. 45, 2026, No. 46, pp 1959-2033.
Published 2026-09-03.
TAC Home