F. William Lawvere

The Category of Categories as a Foundation for Mathematics

In the mathematical development of recent decades one sees clearly the rise of the conviction that the relevant properties of mathematical objects are those which can be stated in terms of their abstract structure rather than in terms of the elements which the objects were thought to be made of. The question thus naturally arises whether one can give a foundation for mathematics which expresses wholeheartedly this conviction concerning what mathematics is about, and in particular in which classes and membership in classes do not play any role. Here by “foundation” we mean a single system of first-order axioms in which all usual mathematical objects can be defined and all their usual properties proved. A foundation of the sort we have in mind would seemingly be much more natural and readily-useable than the classical one when developing such subjects as algebraic topology, functional analysis, model theory of general algebraic systems, etc. Clearly any such foundation would have to reckon with the Eilenberg-MacLane theory of categories and functors. The author believes, in fact, that the most reasonable way to arrive at a foundation meeting these requirements is simply to write down axioms descriptive of properties which the intuitively-conceived category of all categories has until an intuitively-adequate list is attained; that is essentially how the theory described below was arrived at. Various metatheorems should of course then be proved to help justify the feeling of adequacy.

Continuously Variable Sets; Algebraic Geometry = Geometric Logic

Publisher Summary The (elementary) theory of topoi is the basis for the study of continuously variable structures as classical set theory is the basis for the study of constant structures. Sheaves on a coherent topological space also occur in algebraic geometry. This chapter illustrates that algebraic geometry is equal to geometric logic. Each can be transformed into the other on the basis of the fact that both are the studies of continuous maps between coherent topoi. This claim metaphysically ignores the dominating aspect in algebraic geometry of calculations in linear algebra. Logic should be regarded as including the formalism of closed categories, whereas the form of the linear algebra calculations in geometry is that of (abelian and) closed categories.

Variable Quantities and Variable Structures in Topoi

Publisher Summary This chapter discusses variable quantities and variable structures in topoi. The chapter presents the conceptual basis for topoi in mathematical experience with variable sets and discusses a formal theory of variable abstract sets as a relativized foundation for geometry and analysis. It also focuses on sheaves of continuous maps. The chapter presents two aspects of sheaf theory not yet sufficiently incorporated into general topoi theory. There has long been in geometry and differential equations the idea that the category of families of spaces smoothly parametrized by a given space X is similar in many respects to the category of spaces itself. Traditionally, the set theory has emphasized the constancy of sets, and both nonstandard analysis and forcing method involve passing from a system of supposedly constant sets to a new system.

Foundations and Applications: Axiomatization and Education

Foundations and Applications depend ultimately for their existence on each other. The main links between them are education and the axiomatic method. Those links can be strengthened with the help of a categorical method which was concentrated forty years ago by Cartier, Grothendieck, Isbell, Kan, and Yoneda. I extended that method to extract some essential features of the category of categories in 1965, and I apply it here in section 3 to sketch a similar foundation within the smooth categories which provide the setting for the mathematics of change. The possibility that other methods may be needed to clarify a contradiction introduced by Cantor, now embedded in mathematical practice, is discussed in section 5.

Unity and Identity of Opposites in Calculus and Physics

A significant fraction of dialectical philosophy can be modeled mathematically through the use of “cylinders” (diagrams of shape Δ) in a category, wherein the two identical subobjects (united by the third map in the diagram) are “opposite”. In a bicategory, oppositeness can be very effectively characterized in terms of adjointness, but even in an ordinary category it may sometimes be given a useful definition. For example, an effective basis for teaching calculus is a ringed category satisfying the Hadamard-Marx property. The description in engineering mechanics of continuous bodies that can undergo cracking is clarified by an example involving lattices, raising a new questions about the foundations of topology.

ALGEBRAIC THEORIES, ALGEBRAIC CATEGORIES, AND ALGEBRAIC FUNCTORS

Publisher Summary This chapter discusses algebraic theories, algebraic categories, and algebraic functors. The theory of categories and functors extends to the various fields of mathematics a methodological injunction. There are at least four distinct levels in general algebra where maps are to be seen: (1) Homomorphisms between algebraic structures of a given equational type. This suggests the notion of “algebraic category.” (2) “Equational” interpretations between algebraic theories. (3) “Algebraic” functors between algebraic categories induced by the interpretations. (4) The semantical assignment of algebraic categories to algebraic theories. This assignment is a functor which, we discover, also has an adjoint when properly construed. By a mapping between algebraic theories one will understand a functor that preserves products.

Comments on the Development of Topos Theory

Summarizing several threads in the development of the Elementary Theory of Toposes in its first 30 years 1970-2000, this historical article prepares the reader for later publication such as Johnstone’s Elephant (2002) and for the author’s own steps toward an improved foundation for algebraic geometry in the Grothendieck spirit, but using the tools of categorical logic and taking up the theme of axiomatic cohesion.

Linearization of graphic toposes via Coxeter groups

Abstract In an associative algebra over a field K of characteristic not 2, those idempotent elements a , for which the inner derivation [−, a ] is also idempotent, form a monoid M satisfying the graphic identity aba = ab . In case K has three elements and M is such a graphic monoid, then the category of K -vector spaces in the topos of M -sets is a full exact subcategory of the vector spaces in the Boolean topos of G -sets, where G is a crystallographic Coxeter group which measures equality of levels in the category of M -sets.

Categorical algebra for continuum micro physics

Using the setting of a topos equipped with a specified infinitesimal time-interval, we try in part I to clarify the idea of lawful motions as morphisms in a category whose objects are laws of motion on state spaces, and in part II to develop specific relations between states, bodies and particles. A very general scheme to make mass distributions yield notions of inertia and hence of force is discussed. Part III concerns a special notion of a body having just one point, yet containing rich microstructure; such a body is placed in space and, in general, treated like any other body. In part IV some detailed homogeneous and quadratic examples are defined.

Euler’s Continuum Functorially Vindicated

Contrary to common opinion, the question “what is the continuum?” does not have a final answer (Bell, 2005), the immortal work of Dedekind notwithstanding. There is a deeper answer implicit in an observation of Euler. Although it has often been dismissed as naive, we can use the precision of the theory of categories to reveal Euler’s observation to be an appropriate foundation for smooth and analytic geometry and analysis.