! and ?: Storage as tensorial strength

by R.F. Blute, J.R.B. Cockett, and R.A.G. Seely

ABSTRACT: We continue our study of the negation-free structure of multiplicative linear logic, as represented by the structure of weakly distributive categories, to consider the ``exponentials'' ! and ? in the weakly distributive context. In addition to the usual triple and cotriple structure that one would expect on each of the two operators, there must be some connection between them, to replace the de Morgan relationship found in the linear logic context: that turns out to be the notion of tensorial strength. We analyze coherence for this situation via the usual proof nets for linear logic with exponentials, due to Danos (with some minor modifications due to the lack of a negation-induced duality, and with some extra net rewrites needed for the coherence).