The Caus[C] construction takes a base category of unnormalised processes (e.g. stochastic, quantum, or affine maps) and applies double-glueing with an orthogonality to restrict to deterministic/normalisation-preserving processes, lifting the monoidal product of C to many products representing different causal structures. Previous work by the authors has shown such categories are models of MLL and, later, BV. In this follow-on work, we devise a sound and complete logic for compatibility of causal structures with respect to composition and show it is independent of both the base category and the local system types (beyond the identification of first-order systems). This logic is a conservative extension of pomset logic with additional directed axioms, which themselves can be faithfully encoded back into pomset. Using this, we can relate a known separating statement between pomset and BV to a standard result of process matrices in quantum causality.

Joint work with Aleks Kissinger.