Some axiomatic results in synthetic domain theory

Abstract

Working in a model of intuitionistic higher order logic, a topos, we start by taking a dominance, a subobject S of the subobject classifier W satisfying some axioms. These axioms imply that we have a monad L on the topos, also called the lifting monad, which classifies S-partial maps. L has both an initial algebra I and a final coalgebra F; an object X of the topos is complete iff it is orthogonal to the canonical map I → F. The talk presented a new proof for Jibladze's formula for L, and under the additional assumptions that S is complete and ¬¬-separated, it was proved that for a complete algebra for the monad L, every function has a fixed point. Moreover, the complete regular S-posets are closed under the functor L.The material of the talk is in the paper ‘Axioms and (counter)examples in synthetic domain theory, Ann. Pure Appl. Logic 104 (2000) 233-278’ by J. van Oosten and A.K. Simpson.