This chapter provides examples of the central concepts of the book: initial algebras and terminal coalgebras. These are mainly for polynomial endofunctors on the category of sets, but also for such functors on sorted sets, nominal sets, and presheaves. We discuss connections to induction and recursion and to their duals, coinduction and corecursion, and also to bisimulation. We present Lambek’s Lemma, a result used throughout the book.

This chapter presents simple and reachable coalgebras and constructions of the simple quotient of a coalgebra and the reachable part of a pointed one. It introduces well-pointed coalgebras: those which are both reachable and simple. Well-pointed coalgebras constitute a coalgebraic formulation of minimality of state-based systems. For set functors preserving intersections, we prove that the terminal coalgebra is formed by all well-pointed coalgebras, and the initial algebra by all well-founded, well-pointed coalgebras (both considered up to isomorphism) with canonical structures.

The current chapter addresses one of the most controversial concepts in psychology due to the challenge it poses to study from a methodological perspective: the mind. The chapter’s objective is to inform the reader that when accessing digital services, users come with preformed expectations shaped into mental or neural representations, a result of their previous experiences with the service or from what they have learned through social interactions with others. In the decision-making process, where users determine whether to use a digital service or others, they base their choice on a cognitive unit comprised of different mental representations that link a behavior to a reward. The behavior that is decided upon (which would include the use of the digital service) generates a set of expectations, which will be compared to the actual experience. The outcome of this expectation-reality comparison will elicit an emotional response that will become associated with the digital service, altering its value for the user. The congruence between expectation and reality is a fundamental requirement for a cognitively ergonomic design of digital services. Therefore, digital behavior designers must ensure that user expectations align with the reality of interacting with the digital tool. Designing user expectations should be one of the focal points in the design of digital behaviors, as a mismatch between expectation and reality can produce aversive emotions that may lead to the abandonment of the service.

This chapter takes the iterative construction of initial algebras into the transfinite, generalizing work in Chapters 2 and 4. It begins with a brief presentation of ordinals, cardinals, regular cardinals, and Zermelo’s Theorem: Monotone functions on chain-complete posets have least fixed points obtainable by iteration. When a category has colimits of chains, if an endofunctor preserves colimits of chains of some ordinal length, then the initial-algebra chain converges in the same number of steps. We discuss the precise length of that iterative construction. We introduce the concept of smooth monomorphisms, providing a relation between iteration inside a subobject poset and in the ambient category. We prove the Initial Algebra Theorem: Under natural assumptions related to smoothness, the existence of a pre-fixed point of an endofunctor guarantees the existence of an initial algebra.

This chapter discusses terminal coalgebras obtained by methods other than the finitary iteration that we saw in Chapter 3. One way is by taking a quotient of a weakly terminal coalgebra. Another is to use Worrell’s Theorem: the terminal coalgebra of a finitary set functor is obtainable as a limit, using a doubled form of infinite iteration. The chapter also contains a number of presentations of the terminal coalgebra of the finite power-set functor on sets and of the first infinite limit of its terminal-coalgebra chain.

This chapter presents the limit-colimit coincidence in categories enriched either in complete partial orders or in complete metric spaces. This chapter thus works in settings where one has a theory of approximations of objects, either as joins of $\omega$-chains or as limits of Cauchy sequences, and with endofunctors preserving this structure. There are some additional requirements, and we discuss examples. In the settings which do satisfy those requirements, the initial algebra and the terminal coalgebras exist and their structures are inverses, giving what is known as a canonical fixed point (a limit-colimit coincidence). We recover some known results on this topic due to Smyth and Plotkin in the ordered setting and to America and Rutten in the metric setting. We also discuss applications to solving domain equations.

We motivate the book based on categorical formulations of recursion and induction. We also discuss the background that readers should have and preview many of the topics in the book.

This appendix summarizes the results in different parts of the book which give initial algebras, terminal coalgebras, and canonical fixed points.

An essential aspect of designing digital behaviors involves identifying, selecting, and properly designing the reinforcers within a digital space. These reinforcers function to increase the usage of digital services, aiming to achieve corporate objectives through habit formation. Reinforcers can be administered after the user performs the desired behavior; they can be delivered immediately, delayed by seconds or days, and may be administered directly by the technological device or indirectly by the natural environment. Identifying all possible reinforcers before the development of the digital service (reinforcer matrix) can impact the design or development of the service itself. The implementation of reinforcers should be prioritized, finding those that provide the most value in satisfying user needs. This is why identifying, selecting, and designing reinforcers is essential as a preliminary step to the service’s design and development. Once the various reinforcers are detected, it must be considered under what reinforcement schedule they will be administered in the different user interactions with the digital service. Most will be through a continuous reinforcement program, but many others may function better under an intermittent reinforcement program. In conclusion, once the appropriate reinforcers are found, the effects they might have in the environment and long term must be studied to minimize potential adverse effects that could arise in users and the business itself. Only when they are properly identified and selected, does the design of the signals that will indicate to the user the possibility of obtaining them begin. These signals are conceptualized as discriminative stimuli or deltas, and they form the bridge between the design of digital behavior and the design of the user experience.

In the previous chapters, the theoretical foundations upon which the design of digital behaviors is based were presented. This chapter proposes a methodology to organize all this knowledge into a valid process for the design and development of Digital Operant Boxes, or equivalently, digital services. The aim of designing digital behaviors is the creation of behavioral blueprints of the user's potential interactions with the digital service, in such a way that reflects their personal, historical, and digital characteristics. To achieve this objective, this chapter presents a methodology “Digital Behavioral Design” which consists of two main phases with several stages in each: (1) Digital Behavior Analysis and (2) Drives and Operants Design. The Digital Behavior Analysis comprises the following stages: “Goal-Directed Behavior Design” and “Digital Behavioral Map”. This phase focuses on the molar behaviors of the user when using the digital service and allows the designer to present an initial approximation of the service’s behavioral blueprints. The second phase consists of 5 stages: “Behavioral Profiling”, “Reinforcer Matrix”, “Behavioral Competition Analysis”, “Hierarchy Map of Reinforcers”, and “Digital Blueprint”. These stages enable the designer to optimize the drives and reinforcers identified in the first phase and to finely tune the digital service. Therefore, “Digital Behavioral Design” becomes a tool for the designer of digital behaviors that will allow them to apply knowledge from behavioral and cognitive sciences to the design of digital services.

Various cognitive theories indicate how the brain and technology have interacted with each other in an iterative and progressive manner to shape human cognition. Technologies are cultural tools that emerged as a human response to address specific needs. These technologies have allowed us to overcome various ecological, social, and cultural challenges that have impacted the phylogenetic development of higher cognitive abilities that have elevated Homo sapiens above other species. In the digital age, technologies such as the internet, smartphones, and the various software applications that derive from them play a fundamental role in how we relate to ourselves and society. Understanding how humans interact with these technologies, and the effect they have on altering brain architecture, is essential for designing and developing better tools. This chapter summarizes the key findings that explain the consequences of using these technologies on our development and how behavior, through these means, has given rise to digital behavior. Digital behavior is the compendium of interactions and their consequences that occur on the individual when using a digital service. The design of digital behaviors can be described as a new sub-field of Human Factors and Engineering Psychology, with habit formation and need satisfaction serving as the main epistemological core of digital behavior design. The design of digital behaviors is a necessary discipline that can enhance user engagement with these technologies by improving cognitive ergonomics, thereby more effectively addressing the needs that users bring to these types of services.

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.