Fixpoint operators for 2-categorical structures

Fixpoint operators are tools to reason on recursive programs and data types obtained by induction (e.g. lists, trees) or coinduction (e.g. streams). They were given a categorical treatment with the notion of categories with fixpoints. A theorem by Plotkin and Simpson characterizes existence and uniqueness of fixpoint operators for categories satisfying some conditions on bifree algebras and recovers the standard examples of the category Cppo ($\omega$-complete pointed partial orders and continuous functions) in domain theory and the relational model in linear logic. We present a categorification of this result and develop the theory of 2-categorical fixpoint operators where the 2-dimensional framework allows to model the execution steps for languages with (co)inductive principles. We recover the standard categorical constructions of initial algebras and final coalgebras for endofunctors as well as fixpoints of generalized species and polynomial functors.