### Abstract

Language | English |
---|---|

Pages | 239-254 |

Number of pages | 16 |

Journal | Tbilisi Mathematical Journal |

Volume | 10 |

Issue number | 3 |

DOIs | |

Status | Published - 2017 |

### Fingerprint

### Keywords

- enrichment
- internal category
- extensivity
- bicategories
- spans
- matrices

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Enriched and internal categories: an extensive relationship.** / Power, Anthony; Cottrell, Thomas; Fujii, Soichiro.

Research output: Contribution to journal › Article

*Tbilisi Mathematical Journal*, vol. 10, no. 3, pp. 239-254. DOI: 10.1515/tmj-2017-0111

}

TY - JOUR

T1 - Enriched and internal categories: an extensive relationship

AU - Power,Anthony

AU - Cottrell,Thomas

AU - Fujii,Soichiro

PY - 2017

Y1 - 2017

N2 - We consider an extant infinitary variant of Lawvere’s deﬁnition of extensivity of a category V. In the presence of cartesian closedness and ﬁnite limits in V, we give two characterisations of the condition in terms of a biequivalence between the bicategory of matrices over V and the bicategory of spans over discrete objects in V. Using the condition, we prove that V-Cat and the category Catd(V) of internal categories in V with a discrete object of objects are equivalent. Our leading example has V = Cat, making V-Cat the category of all small 2-categories and Catd(V) the category of small double categories with discrete category of objects. We further show that if V is extensive, then so are V-Cat and Cat(V), allowing the process to iterate.

AB - We consider an extant infinitary variant of Lawvere’s deﬁnition of extensivity of a category V. In the presence of cartesian closedness and ﬁnite limits in V, we give two characterisations of the condition in terms of a biequivalence between the bicategory of matrices over V and the bicategory of spans over discrete objects in V. Using the condition, we prove that V-Cat and the category Catd(V) of internal categories in V with a discrete object of objects are equivalent. Our leading example has V = Cat, making V-Cat the category of all small 2-categories and Catd(V) the category of small double categories with discrete category of objects. We further show that if V is extensive, then so are V-Cat and Cat(V), allowing the process to iterate.

KW - enrichment

KW - internal category

KW - extensivity

KW - bicategories

KW - spans

KW - matrices

U2 - 10.1515/tmj-2017-0111

DO - 10.1515/tmj-2017-0111

M3 - Article

VL - 10

SP - 239

EP - 254

JO - Tbilisi Mathematical Journal

T2 - Tbilisi Mathematical Journal

JF - Tbilisi Mathematical Journal

SN - 1512-0139

IS - 3

ER -