Injective Objects and Fibered Codensity Liftings

Functor lifting along a fibration is used for several different purposes in computer science. In the theory of coalgebras, it is used to define coinductive predicates, such as simulation preorder and bisimilarity. Codensity lifting is a scheme to obtain a functor lifting along a fibration. It generalizes a few previous lifting schemes including the Kantorovich lifting. In this paper, we seek a property of functor lifting called fiberedness. Hinted by a known result for Kantorovich lifting, we identify a sufficient condition for a codensity lifting to be fibered. We see that this condition applies to many examples that have been studied. As an application, we derive some results on bisimilarity-like notions.