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Abstract
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.
Original language | English |
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Pages (from-to) | 239-254 |
Number of pages | 16 |
Journal | Tbilisi Mathematical Journal |
Volume | 10 |
Issue number | 3 |
DOIs | |
Publication status | Published - 1 Jun 2017 |
Keywords
- enrichment
- internal category
- extensivity
- bicategories
- spans
- matrices
ASJC Scopus subject areas
- General Mathematics
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Dive into the research topics of 'Enriched and internal categories: an extensive relationship'. Together they form a unique fingerprint.Projects
- 1 Finished
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Higher Category Theoretic Structure of Programming Semantics
Power, J. (PI)
11/11/16 → 11/11/18
Project: Research council