Well quasi orders in a categorical setting

Abstract

This article describes well quasi orders as a category, focusing on limits and colimits. In particular, while quasi orders with monotone maps form a category which is finitely complete, finitely cocomplete, and with exponentiation, the full subcategory of well quasi orders is finitely complete and cocomplete, but with no exponentiation. It is interesting to notice how finite antichains and finite proper descending chains interact to induce this structure in the category: in fact, the full subcategory of quasi orders with finite antichains has finite colimits but no products, while the full subcategory of well founded quasi orders has finite limits but no coequalisers. Moreover, the article characterises when exponential objects exist in the category of well quasi orders and well founded quasi orders. This completes the systematic description of the fundamental constructions in the categories of quasi orders, well founded quasi orders, quasi orders with finite antichains, and well quasi orders.

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Acknowledgements

Thiswork has been supported by the project Correctness byConstruction (CORCON), EU 7 th framework programme, Grant No. PIRSES-GA-2013-612638, European Union Research Agency, and by the project Abstract Mathematics for Actual Computation: Hilbert’s Program in the 21st Century, John Templeton Foundation.

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Correspondence to Marco Benini.

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Benini, M., Bonacina, R. Well quasi orders in a categorical setting. Arch. Math. Logic 58, 501–526 (2019). https://doi.org/10.1007/s00153-018-0649-9

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Keywords

  • Well quasi order
  • Well founded quasi order
  • Preorder categories
  • Exponentiation

Mathematics Subject Classification

  • 06A11
  • 06A75
  • 03G10
  • 18C10