This paper describes a family of logics whose categorical semantics is based on functors
with structure rather than on categories with structure. This allows the consideration of
logics that contain possibly distinct logical subsystems whose interactions are mediated by
functorial mappings. For example, within one unified framework, we shall be able to handle
logics as diverse as modal logic, ordinary linear logic, and the ‘noncommutative logic’ of
Abrusci and Ruet, a variant of linear logic that has both commutative and noncommutative
connectives.
Although this paper will not consider in depth the categorical basis of this approach to
logic, preferring instead to emphasise the syntactic novelties that it generates in the logic, we
shall focus on the particular case when the logics are based on a linear functor, in order to
give a definite presentation of these ideas. However, it will be clear that this approach to
logic has considerable generality.