Conceptual Mathematics

In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of ‘category’ for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.

Table of Contents

SESSION 1 Galileo and multiplication of objects

PART I The category of sets

ARTICLE I Sets, maps, composition. A first example of a category

SESSION 2 Sets, maps and composition

SESSION 3 Composing maps and counting maps

PART II The algebra of composition

ARTICLE II Isomorphisms

SESSION 4 Division of maps: Isomorphisms

SESSION 5 Division of maps: Sections and retractions

SESSION 6 Two general aspects or uses of maps

SESSION 7 Isomorphisms and coordinates

SESSION 8 Pictures of a map making its features evident

SESSION 9 Ret racts and idempotents

Quiz

SESSION 10 Brouwers theorems

PART III Categories of structured sets

ARTICLE Ill Examples of categories

SESSION 11 Ascending to categories of richer st ructures

SESSION 12 Categories of diagrams

SESSION 13 Monoids

SESSION 14 Maps preserve positive properties

SESSION 15 Objectification of properties in dynamical systems

SESSION 16 Idempotents, involut ions, and graphs

SESSION 17 Some uses of graphs

Test 2

SESSION 18 Review of Test 2

PART IV Elementary universal mapping properties

ARTICLE IV Universal mapping properties

SESSION 19 Terminal objects

SESSION 20 Points of an object

SESSION 21 Products in categories

SESSION 22 Universal mapping properties. Incidence relations

SESSION 34 Group theory and the number of types of connected o bjects

SESSION 35 Constants, codiscrete objects, and many connected o bjects

PPENDICES Toward Further Studies

APPENDIX I Geometry of figures and algebra of functions

APPENDIX II Adjoint functors with examples from graphs and dynamical systems

APPENDIX Ill The emergence of category theory within mathematics

APPENDIX IV Annotated Bibliography

About the Authors

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