### Abstract

We give a number of equivalent conditions for a topos to be homotopically trivial and then relate these conditions to the logic of the topos. This is accomplished by constructing a family of intervals that can detect complemented, regular subobjects of the terminals. It follows that these conditions generally are weaker than the Stone condition but are equivalent to it if they hold locally. As a consequence we obtain an extension of Johnstone’s list of conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is no nontrivial homotopy theory in the category of sets is equivalent to the fact, among others, that maximal ideals in commutative rings are prime. Moreover, any topos has a ‘best approximation’ by a locally homotopically trivial topos.

Original language | English (US) |
---|---|

Pages (from-to) | 171-182 |

Number of pages | 12 |

Journal | Pacific Journal of Mathematics |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1984 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*110*(1), 171-182. https://doi.org/10.2140/pjm.1984.110.171

**Homotopically trivial toposes.** / Mielke, Marvin.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 110, no. 1, pp. 171-182. https://doi.org/10.2140/pjm.1984.110.171

}

TY - JOUR

T1 - Homotopically trivial toposes

AU - Mielke, Marvin

PY - 1984/1/1

Y1 - 1984/1/1

N2 - We give a number of equivalent conditions for a topos to be homotopically trivial and then relate these conditions to the logic of the topos. This is accomplished by constructing a family of intervals that can detect complemented, regular subobjects of the terminals. It follows that these conditions generally are weaker than the Stone condition but are equivalent to it if they hold locally. As a consequence we obtain an extension of Johnstone’s list of conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is no nontrivial homotopy theory in the category of sets is equivalent to the fact, among others, that maximal ideals in commutative rings are prime. Moreover, any topos has a ‘best approximation’ by a locally homotopically trivial topos.

AB - We give a number of equivalent conditions for a topos to be homotopically trivial and then relate these conditions to the logic of the topos. This is accomplished by constructing a family of intervals that can detect complemented, regular subobjects of the terminals. It follows that these conditions generally are weaker than the Stone condition but are equivalent to it if they hold locally. As a consequence we obtain an extension of Johnstone’s list of conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is no nontrivial homotopy theory in the category of sets is equivalent to the fact, among others, that maximal ideals in commutative rings are prime. Moreover, any topos has a ‘best approximation’ by a locally homotopically trivial topos.

UR - http://www.scopus.com/inward/record.url?scp=84972584469&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972584469&partnerID=8YFLogxK

U2 - 10.2140/pjm.1984.110.171

DO - 10.2140/pjm.1984.110.171

M3 - Article

AN - SCOPUS:84972584469

VL - 110

SP - 171

EP - 182

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -