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.

This chapter highlights how the pursuit of pleasure, foundational concepts in the philosophy of Epicurus, continue to be essential pillars in the modern understanding of human behavior. These principles are expanded upon by incorporating learning theories formulated by Edward Lee Thorndike, specifically stimulus-response association and the Law of Effect, which posits that actions resulting in pleasure are likely to be repeated, thereby solidifying our understanding of habit formation. Under this paradigm, the influence of gratifying and aversive experiences on our learning and behavior is detailed, emphasizing their central role in the digital age. In particular, it explores how gratifying interactions with mobile devices promote habit formation. Additionally, emerging evidence supporting the concept of the ‘hedonic brain’ is examined, reflecting a neural predisposition towards maximizing pleasure and minimizing pain, and highlighting the importance of dopaminergic brain structures in the storage of gratifying experiences, which will favor their future repetition. The chapter also addresses the mechanisms of positive and negative reinforcement and how these manifest in our interaction with digital technology, focusing on how the digital age has facilitated the attainment of rewards. Finally, the functional analysis of behavior and operant conditioning by Burrhus Frederic Skinner is discussed, illustrating how our behaviors are shaped by their consequences, a principle that is being extensively exploited by technology and digital services.