KZ-monadic categories and their logic

Abstract

Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic for Kan-injectivity consequences and prove that in "reasonable" categories (such as Pos or Top_0) it is also complete for every set H of squares.