# Conceptual Mathematics: A First Introduction to Categories

Cambridge University Press, Oct 9, 1997 - Mathematics - 358 pages
The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study.

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### Contents

 Galileo and multiplication of objects 3 The category of sets 11 Definition of category 21 Sessions Composing maps and counting maps 31 The algebra of composition 37 Special properties a map may have 59 Isomorphisms 60 Sections and retractions 68
 Test 2 204 Elementary universal mapping properties 211 Terminal objects 225 Products in categories 236 Universal mapping properties and incidence relations 245 More on universal mapping properties 254 Uniqueness of products and definition of sum 261 Labelings and products of graphs 269

 Two general aspects or uses of maps 81 Pictures of a map making its features evident 91 Retracts and idempotents 99 Quiz 108 Composition of opposed maps 114 Brouwers theorems 120 Categories of structured sets 133 Ascending to categories of richer structures 152 Categories of diagrams 161 Maps preserve positive properties 170 Idempotents involutions and graphs 187 Some uses of graphs 196
 Distributive categories and linear categories 276 Examples of universal constructions 284 The category of pointed sets 295 TestS 301 Higher universal mapping properties 311 Exponentiation 320 Map object versus product 328 Subobjects logic and truth 335 Toposes 344 Index 353 Copyright

### About the author (1997)

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.