Conceptual Mathematics: A First Introduction to Categories
内容简介
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.
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作者简介
F. William Lawvere, State University of New York, Buffalo
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, State University of New York, Buffalo
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.
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目录
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.
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