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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., Comment: 35 pages

This paper studies presentations of the Sierpinski gasket as a final coalgebra for functors on several categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, a second uses Lipschitz maps, and a third uses continuous maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. We prove that the Sierpinski gasket itself is the final coalgebra of a naturally-occurring functor in the continuous setting. In the short setting, the final coalgebra exists but it is better described as the completion of the initial algebra, and this is not isomorphic to the Sierpinski gasket. In the Lipschitz setting, the final coalgebra does not exist. We determine the initial algebras in all three settings as well., Comment: 21 pages