### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Multiplicity of a space over another space.** / Taniyama, Kouki.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 64, no. 3, pp. 823-849. https://doi.org/10.2969/jmsj/06430823

}

TY - JOUR

T1 - Multiplicity of a space over another space

AU - Taniyama, Kouki

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Category

KW - Group

KW - Knot

KW - Module

KW - Multiplicity

KW - Topological space

UR - http://www.scopus.com/inward/record.url?scp=84866921775&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866921775&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06430823

DO - 10.2969/jmsj/06430823

M3 - Article

VL - 64

SP - 823

EP - 849

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -