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.

The chapter analyses disclosure obligations of environmental and social sustainability risks that apply to companies in light of the growing importance to disclose sustainability risks. In doing so, it discusses the potential cross-border strategies for countries to develop international standards to support global convergence. It considers the international developments justifying the rationale for sustainability-related disclosures along with a discussion of the three models of cross-border disclosure regulation: (i) the home state approach, (ii) the host state approach and (iii) the equivalence approach. The chapter argues that the EU Corporate Sustainability Reporting Directive (CSRD) (2022) has adopted a mix-and-match model between the host state approach and the equivalence approach. Our analysis emphasises the extraterritoriality of EU sustainability disclosure regulation and compares it with the models followed by the United Kingdom, the United States and Switzerland. The different sustainability disclosure requirements between EU countries and non-EU countries suggests, therefore, that cross-border regulatory coordination is important. The paper recommends a model of ESG disclosure for capital markets that is based on the EU policy of equivalence modified by a recognition of the compliance approaches of certain foreign jurisdictions.

Chapter 6 returns to H. G. Wells to offer a fuller account of this writer’s longstanding fascination with animal experimentation, a practice he supported. Analysis of The Wonderful Visit (1895), The Island of Dr Moreau (1896), short stories, and essays reveal this author’s investment in contemporary scientific debates surrounding the thorny issues surrounding non-human pain introduced in Chapter 5. Despite differences in genre and tone, the selected texts each exploit the uneasy relationship between injury, experience, and expression to raise compelling questions about pain’s purpose and limits. The period’s vivisection debates were an important and productive context for Wells who capitalised on the ambivalence they produced, undermined the generic expectations of writings about the subject, and considered whether literary and linguistic methods could uniquely capture – or even solve – the problem of pain.

Economic inequality is not the only form of inequality in urban contexts. In this chapter, I discuss other forms of marginalization in public spaces. Although my main focus is on social relations among citizens, the state’s control of public spaces is consequential in creating and sustaining structural inequalities that directly or indirectly impact social relations in public spaces. Whether controlling appearance and behavior (particularly for women) or suppressing certain belief systems and lifestyles, these state-imposed restrictions create inequalities that extend well beyond economic inequality in use of space. I argue that discriminatory laws or conventions (especially against women and those whose lifestyles or beliefs are not aligned with the ideals promoted by the state) are translated into unequal power relations in public spaces. This chapter examines how these inequalities impact perceptions of class and culture as social groups interact in public spaces and how public spaces are used to create spaces of being and belonging for marginalized groups.

Chapter 7 begins with the reaction that followed from Price and Paine’s defence of the colonial cause. They agreed with the colonists that to live in dependence on the arbitrary will of someone else is what it means to be a slave, and consequently agreed that the colonists must have a natural right to free themselves from their servitude, if necessary by force. This was the moment when a large number of pro-imperial spokesmen came forward to claim that the colonists and their supporters were failing to understand what it means to speak of possessing or losing one’s liberty. They objected that we are not rendered unfree if we are merely subject to someone else’s will; we are only rendered unfree if we are restrained from acting in some particular way. Generally this definition of liberty has been seen as an invention of the late eighteenth century. But as this chapter shows, it arose out of a long tradition of legal and political argument that originated with Hobbes, Pufendorf and their followers. We already find it present in England in the early eighteenth century, and the pro-imperialist writers now brought it to the forefront of debate.

The etiology of metabolic movement disorders is characterized by marked heterogeneity, including acquired and genetic forms (also known as inborn errors of metabolism). In both cases, metabolic alterations represent a possible pathogenetic mechanism of movement disorders that can present with dystonia, parkinsonism, choreoathetosis, or myoclonus, and can range from hyperacute to chronic forms. These conditions can be classified according to multiple aspects, such as etiology, age, or clinical features’ rapidity of onset. Understanding the underlying pathogenic mechanisms has led to specific treatments for acquired and genetic forms and prompt diagnosis is key to reducing brain damage and improving symptoms. Thus, this chapter offers an overview of these conditions’ main clinical and neuroradiologic features to help clinicians in the diagnostic process.

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.

The eighth chapter pursues the urge among artists to imaginatively reconstruct the original structures that became ruins, and not just of individual buildings but of the whole ancient city. Reconstructions are to be seen in two-dimensional ‘flat’ art (paintings, drawings, watercolours, engravings, panoramas) and in three-dimensional architectural models. These occasionally inspired the erection of modern buildings which realised the reconstructed image. Modern reconstructions employ digital and computer-generated imagery. In the twentieth century three-dimensional models of ancient Rome were constructed, and imaginative visions of Rome were devised for cinema and television.

Frame Semantics is foundational to Construction Grammar in both chronological and conceptual terms. Originally developed by Charles J. Fillmore in the late 1970s to 1980s as a theory of semantics that prioritizes language users’ human experience, it views the meaning of linguistic elements in terms of a network of empirical information, which, in turn, motivates the concept represented by the linguistic elements. The theory laid a rich foundation for a variety of approaches associated with Construction Grammar and remains an intellectual resource for further research developments. This chapter focuses on the seminal ideas of Frame Semantics, further advanced in relation to Construction Grammar and the FrameNet project. After an overview of the theory, a variety of frame concepts (e.g., cognitive frame, interactional frame, and linguistic frame) are discussed. We then turn to how frames can effectively explain grammatical ‘well-formedness’ as illustrated by two case studies that were conducted on the path from Frame Semantics to the establishment of Construction Grammar. The last section discusses implications and prospects for the theory of Frame Semantics.

The US Constitution committed to equality in the Thirteenth, Fourteenth, and Fifteenth Amendments following the Civil War. Legislators and judges quickly confronted the question of what these new provisions might mean for private actors. The Radical Republicans aimed to bring the commitment to equal protection into private spaces, propagating republican discourses about the practical requirements of equal citizenship and the potential duties of private actors. However, the Supreme Court soon reached its own countervailing conclusion that only state actors, not private actors, gained duties from the Reconstruction Amendments. While this latter understanding remained firm, private actors effectively gained obligations to equality under the Civil Rights Act of 1964 and later court decisions working around the initial cabining of constitutional equality. Later debates evince a revival of republican-inflected language and arguments for something like horizontal application, even while the country’s jurists viewed such an extension of rights as basically impossible. Several other episodes in constitutional politics, both at the national and state levels, would continue to revisit this question across a range of issue areas.