## Conceptual Mathematics: A First Introduction to CategoriesThe 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 |

### Other editions - View all

Conceptual Mathematics: A First Introduction to Categories F. W. Lawvere,Stephen Hoel Schanuel No preview available - 1997 |

### 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