Sets for Mathematics, new book available

[wlawvere@buffalo.edu]

   The book
               SETS FOR MATHEMATICS
           by F.W. Lawvere and R.Rosebrugh

which is listed in the Cambridge University Press catalogue, is finally
actually available. 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 =E0 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.  Meanwhile the suggestions, criticisms
and corrections that colleagues may offer are eagerly awaited.
Corrections will be posted on the book home page at

http://www.mta.ca/~rrosebru/setsformath