Modal operators and the formal dual of Birkhoff's completeness theorem
Steve Awodey, Jesse W. Hughes · Mathematical Structures in Computer Science · 2003
We present the dual to Birkhoff's variety theorem in terms of predicates over the carrier of a cofree coalgebra (that is, in terms of ‘coequations’). We then discuss the dual to Birkhoff's completeness theorem, showing how closure under deductive rules dualises to yield two modal operators acting on coequations. We discuss the properties of these operators and show that they commute. We prove as our main result the invariance theorem, which is the formal dual of Birkhoff's completeness theorem.