Modal Operators for Coequations
S. Awodey and J. Hughes
Abstract
We present the dual to Birkhoff's variety theorem in terms of
predicates over the carrier of a cofree coalgebra (i.e. in terms of
``coequations''). We then discuss the dual to Birkhoff's
completeness theorem, showing how closure under deductive rules
dualizes to yield two modal operators acting on coequations. We
discuss the properties of these operators and show that they
commute, and we prove as main result the invariance theorem, which
is the formal dual of Birkhoff's completeness theorem.