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Tbilisi Mathematical Journal

Editor-in-Chief: Inassaridze, Hvedri

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1512-0139
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Enriched and internal categories: an extensive relationship

Thomas Cottrell
  • Corresponding author
  • Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom of Great Britain and Northern Ireland
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Soichiro Fujii / John Power
  • Department of Computer Science, University of Bath, Bath BA2 7AY, United Kingdom of Great Britain and Northern Ireland
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-12-07 | DOI: https://doi.org/10.1515/tmj-2017-0111

Abstract

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

Keywords: Enriched categories; internal categories; extensivity; bicategories; spans; matrices

MSC 2010: 18D05; 18D20; 18D35

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About the article

Received: 0201-09-08

Accepted: 2017-11-08

Published Online: 2017-12-07

Published in Print: 2017-12-20


Citation Information: Tbilisi Mathematical Journal, Volume 10, Issue 3, Pages 239–254, ISSN (Online) 1512-0139, DOI: https://doi.org/10.1515/tmj-2017-0111.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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