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Published online by Cambridge University Press: 30 January 2025
Given an endofunctor F we can form various derived endofunctors whose initial algebras and terminal coalgebras are related to those of F. The most prominent example are coproducts of F with constant functors, yielding free F-algebras, cofree F-coalgebras, and free completely iterative F-algebras. An initial algebra exists for a composite functor FG if and only if it does for GF. We also present Freyd’s Iterated Square Theorem and its converse: A functor F on category with finite coproducts has an initial algebra precisely when FF does. The chapter also studies functors on slice categories and product categories, coproducts of functors, double-algebras, and coproducts of monads.
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