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