F. William Lawvere
Robert Rosebrugh

Published January 2003 by Cambridge University Press .
ISBN: 0521010608  image of cover

From the announcement by the authors:
The main text is based on courses given several times at Buffalo and Sackville for third-year students of mathematics, computer science, and other mathematical sciences. Although more advanced than the book Conceptual Mathematics by Lawvere and Schanuel (which is aimed at total beginners) this text develops from scratch the theory of the category of abstract sets and certain other toposes with examples from elementary algebra, differential equations, and automata theory.

Among the reasons offered in the appendix for developing an explicit foundation is the need to have a basis for studying such works as Eilenberg-Steenrod on algebraic topology and Grothendieck on functional analysis and algebraic geometry. Indeed, the appendix lays down a challenging definition of "foundation" which the book itself can only begin to fulfill.

The basic concepts are treated with detailed explanations and with many examples, both in the text and in exercises. After the basics are available, some old topics can be treated in a unifying contemporary spirit, for example

  1. The standard tools for analyzing an arbitrary map are the induced equivalence relation, co-equivalence relation, graph and cograph (cographs have been very frequently pictured in practice but only rarely recognized explicitly); all four of these are shown to arise inevitably as Kan quantifications, along the two possible interpretations of the generic map as half of the splitting of a generic idempotent.
  2. The so-called "measurable" cardinals can be excluded from a topos by the intuitive demand that space and quantity have a good duality, made explicit à la Isbell via the requirement that there is a fixed automaton such that the monad obtained by double-dualizing into it is the identity.

The authors hope that this work will serve as one of the springboards to the development and teaching of a foundation suitable for twenty-first century mathematics.

1. Abstract sets and mappings
2. Sums, monomorphisms and parts
3. Finite inverse limits
4. Colimits, epimorphisms and the axiom of choice
5. Mapping sets and exponentials
6. Summary of the axioms and an example of variable sets
7. Consequences and uses of exponentials
8. More on power sets
9. Introduction to variable sets
10. Models of additional variation

264 pages, 84 line diagrams, 219 exercises

A sample chapter is available.

See the featured review by John Bell on MathSciNet - requires subscription.

F. W. Lawvere home page
R. Rosebrugh home page