There is a growing concern about the evolution of violent extremism in the digital era. This chapter presents historical progression and current state of how extremists have used digital advancements to increase their reach and influence for their own nefarious purposes. This chapter also discusses the challenges due to encryption and the need for a strategic collaboration and comprehensive whole-of-society approach to combat the threats effectively.

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.

Platform governance and regulation have been salient political issues in Brazil for years, particularly as part of Congress’ response to democratic threats posed by former President Bolsonaro. The question became even more important after the January 8th attempted insurrection in Brasília, which many blame on social media. This includes the newly installed Lula administration. In a letter read on the February 2023 UNESCO “Internet for Trust” global conference, the President, now in his third (non-consecutive) term in office wrote that the attack on the nation’s seats of power was “the culmination of a campaign, initiated much before, and that used, as ammunition, lies and disinformation,” which “was nurtured, organized, and disseminated through several digital platforms and messaging apps.” The new administration has made platform regulation a policy priority, with regulatory and administrative pushes across the board. Brazil has been a battleground where proposals for platform responsibility have been advanced — and disputed.

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.

Fifteen years ago in All Politics is Global, I developed a typological theory of global economic governance, arguing that globalization had not transformed international relations but merely expanded the arenas of contestation to include policy arenas that had previously been the exclusive province of domestic politics. In my model, what truly mattered to global governance was the distribution of preferences among the great powers. When great power coordination was achieved, then effective governance would be the outcome. When great power coordination was not, then global governance would exist in name only. Demands for greater content moderation across platforms have increased as the modern economy has become increasingly data-driven. Can any standards be negotiated at the global level? The likeliest result will be a hypocritical system of “sham governance.” Under this system, a few token agreements might be negotiated at the global level. Even these arrangements, however, will lack enforcement mechanisms and likely be honored only in the breach. The regulatory center of gravity will remain at the national level. Changes at the societal and global levels over the past fifteen years only reinforce the dynamics that lead to such an outcome.

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.