Conceptual Mathematics: A First Introduction to Categories

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Cambridge University Press, Oct 9, 1997 - Mathematics - 358 pages
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In the last fifty years, the use of the notion of 'category' has led to a remarkable unification and simplification of mathematics. Written by two of the best known participants in this development, Conceptual Mathematics is the first book to serve as a skeleton key to mathematics for the general reader or beginning student and as an introduction to categories for computer scientists, logicians, physicists, linguists etc. While the ideas and techniques of basic category theory are useful throughout modern mathematics, this book does not presuppose knowledge of specific fields but rather develops elementary categories such as directed graphs and discrete dynamical systems from the beginning. The fundamental ideas are then illuminated in an engaging way by examples in these categories.
 

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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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