A set functor is an endofunctor on the category of sets. Although the topic of set functors is quite large, there are few if any chapter-length summaries directed to a researcher in the area of this book. This appendix collects the results on set functors that such a person ought to know, including the main preservation properties, such as preservation of weak pullbacks and of finite intersections. It contains the main examples of set functors used in the recent literature and a chart of their preservation properties. Studying monoid-valued functors, it connects the preservation properties of the functor to algebraic properties of the monoid. It presents Trnkova’s modification of a set functor at the empty set needed to obtain a functor preserving all finite intersections.

This appendix summarizes all of the known fixed point theorems used in the book. In addition to the best known results of this type, it also contains Markowsky’s characterization of directed-complete partial orders, Iwamura’s Lemma, and Pataraia’s ordinal-free version of Zermelo’s Theorem (see Chapter 6). It also mentions induction principles related to these fixed point theorems.

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.

This chapter studies results whereby a set functor is lifted to other categories, paying attention to whether the initial algebra and terminal coalgebra structures also lift. For example, given a set functor F having a terminal coalgebra and a lifting on either complete partial orders and complete metric spaces, the terminal coalgebra can be equipped with a canonical order or metric, respectively, so that this yields the terminal coalgebra for the lifting. Initial algebras, however, need not lift from Set to the other categories. We are also interested in specific liftings of F to pseudometric spaces, such as the Kantorovich and Wasserstein liftings. We study extensions to Kleisli categories and liftings to Eilenberg–Moore categories. We present results on coalgebraic trace semantics, and discuss examples such as the classical trace semantics of (probabilistic) labelled transition systems and languages accepted by nominal automata. We also study generalized determinization of coalgebras of functors arising from liftings to Eilenberg–Moore categories, leading to the coalgebraic language semantics. We see many instances of this semantics: the language semantics of non-deterministic weighted, probabilistic, and nominal automata; and also context-free languages.