## Conceptual Mathematics: A First Introduction to CategoriesIn 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 |

### Other editions - View all

Conceptual Mathematics: A First Introduction to Categories F. William Lawvere,Stephen H. Schanuel No preview available - 2009 |

### Common terms and phrases

algebra Alysia arrows assigns associative law automorphism base point binary operation Brouwer's calculate called cartesian closed category category of graphs category of sets CHAD commutes compose composition of maps coproduct corresponding D A N define denoted disk distributive law domain and codomain dots dynamical systems endomap equations exactly one map example Exercise figure of shape finite sets fixed point gives idea idempotent identity laws identity map inclusion map initial object injective internal diagram inverse involution irreflexive isomorphism loop map g map objects map of graphs maps of sets means monomorphism motion multiplication of numbers natural numbers number of elements number of maps Omer pair of maps particular picture precisely proof prove real numbers reflexive graphs satisfy Session Show solution sort source and target space specified subcategory subobject Suppose terminal object universal mapping property universal property