F. William Lawvere is a Professor Emeritus of Mathematics at the State University of New York. He has previously held positions at Reed College, the University of Chicago and the City University of New York, as well as visiting Professorships at other institutions worldwide. At the 1970 International Congress of Mathematicians in Nice, Prof. Lawvere delivered an invited lecture in which he introduced an algebraic version of topos theory which united several previously 'unrelated' areas in geometry and in set theory; over a dozen books, several dozen international meetings, and hundreds of research papers have since appeared, continuing to develop the consequences of that unification. Stephen H. Schanuel is a Professor of Mathematics at the State University of New York at Buffalo. He has previously held positions at Johns Hopkins University, Institute for Advanced Study and Cornell University, as well as lecturing at institutions in Denmark, Switzerland, Germany, Italy, Colombia, Canada, Ireland, and Australia. Best known for Schanuel's Lemma in homological algebra (and related work with Bass on the beginning of algebraic K–theory), and for Schanuel's Conjecture on algebraic independence and the exponential function, his research thus wanders from algebra to number theory to analysis to geometry and topology.

• Foreword; Note to the reader; Preview; Part I. The Category of Sets: 1. Sets, maps, composition; Part II. The Algebra of Composition: 2. Isomorphisms; Part III. Categories of Structured Sets: 3. Examples of categories; Part IV. Elementary Universal Mapping Properties: 4. Universal mapping properties; Part V. Higher Universal Mapping Properties: 5. Map objects; 6. The contravariant parts functor; 7. The components functor; Appendix 1. Geometry of figures and algebra of functions; Appendix 2. Adjoint functors; Appendix 3. The emergence of category theory within mathematics; Appendix 4. Annotated bibliography.

'This text, written by two experts in Category Theory and tried out carefully in courses at SUNY Buffalo, provides a simple and effective first course on conceptual mathematics.' American Mathematical Monthly