## Conceptual Mathematics: A First Introduction to CategoriesIn 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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### 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 |

385 | |

386 | |

### Common terms and phrases

A L B abstract sets adjoint adjoint functors algebra arrows assigns associative law automorphism Brouwer’s calculate called category of graphs category of sets choice problem compose composition of maps corresponding deﬁned deﬁnition denoted disk domain and codomain dots dynamical systems endomap epimorphism equations exactly one map example Exercise Fatima ﬁgure of shape ﬁnd ﬁnite sets ﬁrst ﬁxed point functor g o f give go f idempotent identity laws identity map inclusion map initial object injective internal diagram inverse involution irreﬂexive isomorphism loop map f map g map objects map of graphs maps of sets means monoid monomorphism motion multiplication of numbers natural numbers number of elements number of maps pair of maps particular picture proof prove real numbers reﬂexive graphs right adjoint satisﬁes satisfy section for f Session Show solution sort space speciﬁed subcategory subobject Suppose terminal object unique universal mapping property