### Abstract

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated R-modules and R-linear maps over a principal ideal domain R, and the neighbourhood category of oriented knots in the 3-sphere.

Original language | English |
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Pages (from-to) | 823-849 |

Number of pages | 27 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Oct 5 |

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### Keywords

- Category
- Group
- Knot
- Module
- Multiplicity
- Topological space

### ASJC Scopus subject areas

- Mathematics(all)