Open Access
June 2017 Enriched and internal categories: an extensive relationship
Thomas Cottrell, Soichiro Fujii, John Power
Tbilisi Math. J. 10(3): 239-254 (June 2017). DOI: 10.1515/tmj-2017-0111


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

Funding Statement

We gratefully acknowledge the support of Royal Society grant IE160402. The second author is supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. No data were generated in association with this paper.


Download Citation

Thomas Cottrell. Soichiro Fujii. John Power. "Enriched and internal categories: an extensive relationship." Tbilisi Math. J. 10 (3) 239 - 254, June 2017.


Received: 8 September 2017; Revised: 8 November 2017; Published: June 2017
First available in Project Euclid: 20 April 2018

zbMATH: 06825311
MathSciNet: MR3733876
Digital Object Identifier: 10.1515/tmj-2017-0111

Primary: 18D05
Secondary: 18D20 , 18D35

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

Rights: Copyright © 2017 Tbilisi Centre for Mathematical Sciences

Vol.10 • No. 3 • June 2017
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