This chapter presents initial algebras and terminal coalgebras obtained by the most common method. For the initial algebra, this is by iteration starting from the initial object through the natural numbers. For the terminal coalgebra, this is the dual: the iteration begins with the terminal object. The chapter is mainly concerned with examples drawn from sets, posets, complete partial orders, and metric spaces.

This chapter highlights connections of the book’s topics to structures used in all areas of mathematics. Cantor famously proved that no set can be mapped onto its power set. We present some analogous results for metric spaces and posets. On the category of topological spaces, we consider endofunctors built from the Vietoris endofunctor using products, coproducts, composition, and constant functors restricted for Hausdorff spaces. Every such functor has an initial algebra and a terminal coalgebra. Similar results hold for the Hausdorff functor on (complete) metric spaces. Extending a result of Freyd, we exhibit structures on the unit interval [0, 1] making it a terminal coalgebra of an endofunctor on bipointed metric spaces. The positive irrationals and other subsets of the real line are described as terminal coalgebras or corecursive algebras for some set functors, calling on results from the theory of continued fractions.

This paper summarizes the United States’ legal framework governing Internet “platforms” that publish third-party content. It highlights three key features of U.S. law: the constitutional protections for free speech and press, the statutory immunity provided by 47 U.S.C. § 230 (“Section 230”), and the limits on state regulation of the Internet. It also discusses US efforts to impose mandatory transparency obligations on Internet “platforms.”

Like information disseminated through online platforms, infectious diseases can cross international borders as they track the movement of people (and sometimes animals and goods) and spread globally. Hence, their control and management have major implications for international relations, and international law. Drawing on this analogy, this chapter looks to global health governance to formulate suggestions for the governance of online platforms. Successes in global health governance suggest that the principle of tackling low-hanging fruit first to build trust and momentum towards more challenging goals may extend to online platform governance. Progress beyond the low-hanging fruit appears more challenging: For one, disagreement on the issue of resource allocation in the online platform setting may lead to “outbreaks” of disinformation being relegated to regions of the world that may not be at the top of online platforms’ market priorities lists. Secondly, while there may be wide consensus on the harms of infectious disease outbreaks, the harms from the spread of disinformation are more contested. Relying on national definitions of disinformation would hardly yield coherent international cooperation. Global health governance would thus suggest that an internationally negotiated agreement on standards as it relates to disinformation may be necessary.

Discussing ethics within the highly competitive technology sector can be complex. Although companies outwardly express their commitment to ethical practices, internally, the topic is often regarded as uncomfortable due to its implications for corporate finances. In digital behavior design, ethical consideration begins with defining the various drives and reinforcers that will guide the design process. As a sub-field of Human Factors and Engineering Psychology, digital behavior design can draw from their established Codes of Ethics and Codes of Conduct. However, beyond incorporating the ethical principles that govern Human Factors and psychology, it is also imperative to acknowledge unique ethical principles that highlight the particularities of their application. In seeking these specific ethical norms, it is important to identify the essential desirable value inherent in the professional practice of digital behavior designers. Primarily, the genuine value provided by digital behavior design is utilitarian in nature; that is, it fulfills user needs through satisfaction. From this continuous satisfaction may arise a dependency on these digital services for happiness, leading to problematic online behaviors. Therefore, poor design or the unethical use of discriminative stimuli (nudges) and reinforcers can be highly hazardous for populations with certain psychosocial and neural vulnerabilities. This chapter introduces certain standards to guide ethical and responsible conduct for designers when creating digital services. It also proposes a solution in the form of an algorithm that could be implemented in digital services to detect and support compulsive behaviors.

This chapter is an indispensable part of the framework for the design of digital behaviors, as it lays the foundation upon which user behavior when engaging with digital services can be anticipated. The core principle to grasp is that the probability of a user repeating the use of a digital or virtual service is primarily linked to two variables: satiation of the drive and the emotion generated. The satiation of the drive is elucidated by the drive reduction theory, whereas the proposal of how emotion or excitement selects behavior is explained by Frederick Sheffield’s induced emotion theory. The key point here is the attainment of the reinforcer, which is the element that induces changes both in the reduction of the drive and in the induction of emotion. In relation to this, it has also been suggested to differentiate among drive, motivation, and reinforcer. Drive is an internal state of physiological or psychological origin, acting as a vector of behavior, but it does not energize the behavior per se. This drive triggers a biochemical cascade within the organism that intensifies over time, and its non-satiation progressively energizes behavior, resulting in a subjective experience defined as an emotion, which initiates goal-directed behavior. Finally, a list of different types of drives (survival, sexual, social, and meta-cognitive) has also been offered, indicating the potential needs of users when turning to a digital or virtual service.

In order to manage the issue of diversity of regulatory vision, States may, to some extent, harmonize substantive regulation—eliminating diversity. This is less likely than States determining unilaterally or multilaterally to develop manageable rules of jurisdiction, so that their regulation applies only in limited circumstances. The fullest realization of this “choice of law” solution would involve geoblocking or other technology that divides up regulatory authority according to a specified, and a perhaps agreed, principle. Geoblocking may be costly and ultimately porous, but it would allow different communities to effectuate their different visions of the good in the platform context. To the extent that the principles of jurisdiction are agreed, and are structured to be exclusive, platforms would have the certainty of knowing the requirements under which they must operate in each market. Of course, different communities may remain territorial states, but given the a-territorial nature of the internet, it may be possible for other divisions of authority and responsibility to develop. Cultural affinity, or political perspective, may be more compelling as an organizational principle to some than territorial co-location.

The theme of this chapter is the relation between the initial algebra for a set functor and the terminal coalgebra, assuming that both exist and that the endofunctor is non-trivial. We introduced a notion called pre-continuity. Pre-continuous set functors generalize finitary and continuous set functors. For such functors, the initial algebra and the terminal coalgebra have the same Cauchy completion and the same ideal completion: the $\omega$-iteration of the terminal-coalgebra chain. It follows that for a non-trivial continuous set functor, the terminal coalgebra is the Cauchy completion of the initial algebra.

On October 27, 2022, the Digital Services Act (DSA) was published in the Official Journal of the European Union (EU). The DSA, which has been portrayed as the Europe’s new “Digital Constitution”, sets out a cross-sector regulatory framework for online services and regulates the responsibility of online intermediaries for illegal content. Against this background, this chapters provides a brief overview of recent regulatory developments regarding platform responsibility in the EU. The chapter seeks to add a European perspective to the global debate about platform regulation. Section 3.1 provides an overview of the regulatory framework in the EU and recent legislative developments. Section 3.2 analyses different approaches regarding the enforcement of rules on platform responsibility. Section 3.3 takes a closer look at the regulation of content moderation by digital platforms in the EU. Finally, Section 3.4 adds some observations on the international effects of EU rules on platform responsibility.

While the social media and digital platforms started with an objective to enhance social connectivity and information sharing, they also present a significant challenge in content moderation resulting in spreading disinformation. Disinformation Paradox is a phenomenon where an attempt to regulate harmful content online can inadvertently amplifies it. The social media platforms often serve as breeding grounds for disinformation. This chapter discusses the inherent difficulties in moderating content at a large scale, different responses of these platforms and potential solutions.

This chapter examines China’s approach to platform responsibility for content moderation. It notes that China’s approach is rooted in its overarching goal of public opinion management, which requires platforms to proactively monitor, moderate, and sometimes censor content, especially politically sensitive content. Despite its patchy and iterative approach, China’s platform regulation is consistent and marked by its distinct characteristics, embodied in its defining of illegal and harmful content, its heavy platform obligations, and its strong reliance on administrative enforcement measures. China’s approach reflects its authoritarian nature and the asymmetrical power relations between the government and private platforms. This chapter also provides a nuanced understanding of China’s approach to platform responsibility, including Chinese platforms’ "conditional liability" for tort damages and the regulators’ growing emphasis on user protection and personal information privacy. This chapter includes a case study on TikTok that shows the interplay between the Chinese approach, oversees laws and regulations and the Chinese online platform’s content moderation practices.

The rational fixed point of an endofunctor is a fixed point which is in general different from its initial algebra and its terminal coalgebra. It collects precisely the behaviours of all ‘finite’ coalgebras of a given endofunctor. For sets, they are those with finitely many states. Examples of rational fixed points include regular languages, eventually periodic and rational streams, etc. To study the rational fixed point in categories beyond sets, we discuss locally finitely presentable categories, and we do so in some detail. We characterize the rational fixed point as an initial iterative algebra. The chapter goes into details on many examples, such as rational fixed points in nominal sets. It discusses the rational fixed point and several other fixed points as well, and it summarizes much of what is known about them.

Chapter 6 may be regarded as an extension of Chapter 5, aiming to discuss several important aspects concerning secondary or psychological drives, given that digital services widely consumed by the general population tend to cater to the satiation of one or more of these drives. These drives or needs have emerged within a cultured society that reaches into the human psyche, taking on a significant role in regulating interactions among different members of society. With the advent of social networks, and the continuous interactions we engage in through them, certain psychological drives such as status or control over identity are becoming a powerful mobilizing force for goal-directed behavior. Hence, it has been deemed timely to delve deeper into some specific drives in this regard.