Free PLTL Algebras and A Coalgebraic LTL Extension of Hyperdoctrines

Abstract

In this seminar, I describe the work I had done over the summer of 2025 at the Topos Institute with José Vitor Paiva Miranda de Siqueira. Inspired by the free Boolean/Heyting algebra of a given set, we develop a free-forgetful adjunction between posets and PLTL temporal algebras, where PLTL denotes propositional linear temporal logic. We provide a description of their induced Eilenberg-Moore categories. We describe how this could be used to temporalise systems and logics through hyperdoctrines and connect this to the stream comonad. We end with future research directions, connecting this topic with the cofree comonad of polynomial functors and temporalising doxastic logic.