The problem of finding map objects in a given category is complicated by the fact that often the map object we are looking for does not exist. This difficulty occurs many times in mathematics: we have a problem and we do not even know whether it has a solution. In such cases it is often helpful to pretend that the problem does have a solution, and proceed to calculate anyway! We need an account of

which we will then apply to the problem of determining map objects in the category of sets and in the category of graphs.

We imagine that we have already found the solution to a given problem, and try to deduce consequences from its existence. We ask ourselves: What does this solution imply? In this way we are often able to deduce enough properties of that solution to discover the real way to the solution or to prove that a solution is impossible.

To apply this method there are two parts, both of which are important. The first is to find out as much as possible about the solution one seeks under the assumption that a solution does exist. Usually one proves first a conclusion of the following type: If a solution exists it must be a certain thing. But the thing found may not be a solution. The second part consists in verifying that, indeed, this thing really is a solution to the problem.

We use maps to express extra ‘structure’ on sets, leading to graphs, dynamical systems, and other examples of ‘types of structure.’ We then investigate ‘structure-preserving’ maps.

Since its first introduction over 60 years ago, the concept of category has been increasingly employed in all branches of mathematics, especially in studies where the relationship between different branches is of importance. The categorical ideas arose originally from the study of a relationship between geometry and algebra; the fundamental simplicity of these ideas soon made possible their broader application.

The categorical concepts are latent in elementary mathematics; making them more explicit helps us to go beyond elementary algebra into more advanced mathematical sciences. Before the appearance of the first edition of this book, their simplicity was accessible only through graduate-level textbooks, because the available examples involved topics such as modules and topological spaces.

Our solution to that dilemma was to develop from the basics the concepts of directed graph and of discrete dynamical system, which are mathematical structures of wide importance that are nevertheless accessible to any interested high-school student. As the book progresses, the relationships between those structures exemplify the elementary ideas of category. Rather remarkably, even some detailed features of graphs and of discrete dynamical systems turn out to be shared by other categories that are more continuous, e.g. those whose maps are described by partial differential equations.

Many readers of the first edition have expressed their wish for more detailed indication of the links between the elementary categorical material and more advanced applications.

Balls, spheres, fixed points, and retractions

The Dutch mathematician L.E.J. Brouwer (1881–1966) proved some remarkable theorems about ‘continuous’ maps between familiar objects: circle, disk, solid ball, etc. The setting for these was the ‘category of topological spaces and continuous maps.’ For our purposes it is unnecessary to have any precise description of this category; we will instead eventually list certain facts which we will call ‘axioms’ and deduce conclusions from these axioms. Naturally, the axioms will not be selected at random, but will reflect our experience with ‘cohesive sets’ (sets in which it makes sense to speak of closeness of points) and ‘continuous maps.’ (Roughly, a map f is continuous if f(p) doesn't instantaneously jump from one position to a far away position as we gradually move p. We met this concept in discussing Galileo's idea of a continuous motion of a particle, i.e. a continuous map from an interval of time into space.) There is even an advantage in not specifying our category precisely: our reasoning will apply to any category in which the axioms are true, and there are, in fact, many such categories (‘topological spaces’, ‘smooth spaces’, etc.).

We begin by stating Brouwer's theorems and by trying to see whether our intuition about continuous maps makes them seem plausible. First we describe the Brouwer fixed point theorems.

Undergraduate texts:

J. Bell, A Primer of Infinitesimal Analysis, Cambridge University Press 1998.

Using some basic category theory, the traditional use of nilpotent infinitesimals in the classical analysis and geometry of Euler, Lie, and Cartan is placed on a rigorous footing and applied to numerous traditional calculations of central importance in elementary geometry and engineering. Like Conceptual Mathematics, this book could serve as a detailed guide to the construction of a course at the beginning level.

M. La Palme Reyes, G. E. Reyes, H. Zolfaghari Generic Figures and their Glueings, Polimetrica, 2004.

Several examples of ‘presheaf’ toposes, with particular properties of map spaces and truth value spaces, are discussed in detail, continuing the elaboration of our present Part V and Appendix II beyond the simple graphs and dynamical systems, with ‘windows’ into various mathematical applications.

R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers, 1996.

This was the first text that introduced differential geometry synthetically, using the categorical foundation of that subject as developed by Kock and others.

F. W. Lawvere, R. Rosebrugh, Sets for Mathematics, Cambridge University Press, 2003.

This text explains on a higher level many of the topics from Conceptual Mathematics, with more advanced examples from set theory. Axioms for the category of sets serve as a foundation for the mathematical uses of set theory as a background for topological, algebraic, and analytical structures, their manipulation, and their description by means of logic.

For over a millennium, and until the nineteenth century, the Shariʿa represented a complex set of social, economic, moral, educational, intellectual and cultural practices. It was not just about law. It pervaded social structures so deeply that no ruler could conceive of the possibility of efficiently ruling the population without succumbing to a great extent to the dictates of the Shariʿa order. Involving institutions, groups and processes that resisted, enhanced and affected each other, Shariʿa as practice manifested itself as much in the judicial process as in writing, studying, teaching and documenting. It manifested itself in political representation, and in strategies of resistance against political and other abuses, as well as in cultural categories that meshed into ethical codes and a moral view of the world. It lived and operated in a deeply moral community which it took as granted, for it is a truism that the Shariʿa itself was constructed on the assumption that its audiences and consumers were, all along, moral communities and morally grounded individuals.

The Shariʿa also involved a complex and sophisticated intellectual system in which the jurists and the members of the legal profession were educators and thinkers who, on the one hand, were historians, mystics, theologians, logicians, men of letters and poets, and, on the other, contributed to the forging of a multi-layered set of relations that at times created political truth and ideology while at other times confronting power with its own truth.

Introduction

With the background provided in the previous chapter, we now turn to discuss how the class of legists perpetuated itself and how its evolution intertwined with the interests of the ruler. During the first two or three centuries of Islam, education was largely and deliberately disconnected from politics, being limited to private scholarship which the rulers sought to influence without much success. The story of this chapter is that of the transformation of legal scholarship from a highly independent enterprise to a markedly subordinate system that came to serve the ruler and his administration. However, as we have already intimated, this eventual subordination did not mean that the content of the law and its application was compromised by any political accommodation. In fact, it was the ruler who – from the beginning of Islam until the middle of the nineteenth century – consistently had to bow to the dictates of the Shariʿa and its representatives in governing the populace. As a moral force, and without the coercive tools of a state, the law stood supreme for over a millennium.

Sometime during the late tenth century, the law college, known in Islamic languages as the MADRASA, came into being, exhibiting a tendency to superimpose itself over the study circle, and in the long run changing some if its features.

Methods of legal reform

By 1900, the Shariʿa in the vast majority of Muslim lands had been reduced in scope of application to the area of personal status, including child custody, inheritance, gifts and, to some extent, waqf. In the Malay states and the Indonesian Archipelago, its sphere was even narrower, partly because of the adat which had long prevailed in some of these domains, and partly because of massive Westernization of its contents and form. The present chapter therefore focuses on personal status, following the fortunes (indeed misfortunes) of Islamic law roughly from the end of World War I until the dawn of the twenty-first century.

The Islamic law of personal status was saved from the death blows dealt to other Shariʿa laws (except rituals) by virtue of the fact that it was of no use to the colonial powers as a tool of domination. Even this disinterest was turned into an advantage, however, for colonialist Europe and its academics promoted the idea that the personal law was sacred to Muslims and that, out of sensitivity and respect, colonial powers left it alone. However, once the laws of personal status were culturally marked as such, they were taken as the point of reference for the modern politics of identity. If family law emerged as “the preferential symbol of Islamic identity,” it did so not only because it was built into Muslim knowledge as an area of a sensitive nature, but also because it represented what was taken to be the last fortress of the Shariʿa to survive the ravages of modernization.