Kenney, Toby (2007) *GENERATING FAMILIES IN A TOPOS.* [Preprint]

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## Abstract

A generating family in a category C is a collection of objects {A_i|i\in I} such that if for any subobject Y \rightarrow^m X, every A_i \rightarrow^f X factors through m, then m is an isomorphism – i.e. the functors C(A_i, \_) are collectively conservative. In this paper, we examine some circumstances under which subobjects of 1 form a generating family. Objects for which subobjects of 1 do form a generating family are called partially well-pointed. For a Grothendieck topos, it is well known that subobjects of 1 form a generating family if and only if the topos is localic. For the elementary case, little more is known. The problem is studied in [1], where it is shown that the result is internally true, an equivalent condition is found in the boolean case, and certain preservation properties are shown. We look at two different approaches to the problem, one based on a generalization of projectivity, and the other based on looking at the most extreme sorts of counterexamples.

Item Type: | Preprint |
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Additional Information: | pubdom TRUE |

Uncontrolled Keywords: | AARMS, Topoi, Generating Families, Cogenerators, Semiprojective Objects |

Subjects: | 18-xx Category theory, homological algebra 03-xx Mathematical logic and foundations |

Faculty: | UNSPECIFIED |

Depositing User: | lingyun ye |

Date Deposited: | 14 Jun 2007 |

Last Modified: | 27 Apr 2010 16:37 |

URI: | https://docserver.carma.newcastle.edu.au/id/eprint/360 |

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