Conceptual Mathematics: A First Introduction to Categories

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Cambridge University Press, Oct 9, 1997 - Mathematics - 358 pages
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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
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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.

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