Conceptual Mathematics: A First Introduction to Categories

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Cambridge University Press, 2009 - Mathematics - 390 pages
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In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. 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. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.
  

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low level intro to category theory that is uniquely accessible to undergrads Read full review

Contents

Galileo and multiplication of objects IUJUJUJ
3
Session
4
The category of sets
11
Session
13
Session
16
Definition of category
21
Session
24
Session
30
68
168
Monoids
170
Paths
196
Elementary universal mapping properties
211
Terminal objects
225
Universal mapping properties and incidence relations
245
135
280
Binary operations and diagonal arguments
302

Composing maps and counting maps
31
Session 9
34
The algebra of composition
37
Special properties a map may have
59
Quiz
60
Sections and retractions
68
Two general aspects or uses of maps
81
Two abuses of isomorphisms
89
Retracts and idempotents
99
Comparing infinite sets
106
Composition of opposed maps
114
Session 10
120
Ascending to categories of richer structures
152
Categories of diagrams
161
70
308
Higher universal mapping properties
311
81
315
LIIAUJNH
320
Map object versus product
328
The contravariant parts functor
335
Toposes
348
The Connected Components Functor
358
Constants codiscrete objects and many connected objects
366
Adjoint functors with examples from graphs and dynamical systems
372
The emergence of category theory within mathematics
378
136
385
86
386
Copyright

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About the author (2009)

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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