Continuous Theories

Abstract

In the first three sections of this chapter we consider theories whose hom-sets are equipped with a partial order which is compatible with the theory operations; iteration is defined using least fixed points. An ordered theory is a special kind of 2-theory, one in which there is a vertical morphism f → g iff f ω g. In Section 4, the connection between initiality and the fixed point properties of iteration is examined in the context of 2-theories. We consider the properties of initial f-algebras, for horizontal morphisms f in a 2-theory. Many properties of iteration theories hold when all such initial algebras exist. In the last section, some of the constructions for ω-continuous ordered theories are generalized to ω-continuous 2-theories. In particular, it is shown how any ω-continuous 2-theory determines an iteration theory.