On Equality of Objects in Categories in Constructive Type Theory

Author Erik Palmgren

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Erik Palmgren
  • Department of Mathematics, Stockholm University, Stockholm, Sweden

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Erik Palmgren. On Equality of Objects in Categories in Constructive Type Theory. In 23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 7:1-7:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In this note we remark on the problem of equality of objects in categories formalized in Martin-Löf's constructive type theory. A standard notion of category in this system is E-category, where no such equality is specified. The main observation here is that there is no general extension of E-categories to categories with equality on objects, unless the principle Uniqueness of Identity Proofs (UIP) holds. We also introduce the notion of an H-category with equality on objects, which makes it easy to compare to the notion of univalent category proposed for Univalent Type Theory by Ahrens, Kapulkin and Shulman.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • type theory
  • formalization
  • category theory
  • setoids


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  1. Peter Aczel. Galois: a theory development project. Report for the 1993 Turin meeting on the Representation of Mathematics in Logical Frameworks, 1995. URL: http://www.cs.man.ac.uk/~petera/papers.html.
  2. Benedikt Ahrens, Krzysztof Kapulkin, and Michael Shulman. Univalent categories and the Rezk completion. Math. Structures Comput. Sci., 25(5):1010-1039, 2015. Google Scholar
  3. Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in type theory. Journal of Functional Programming, 13:261-293, 2003. Google Scholar
  4. Errett Bishop. Foundations of constructive analysis. McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. Google Scholar
  5. Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. Google Scholar
  6. Erik Palmgren. Proof-relevance of families of setoids and identity in type theory. Arch. Math. Logic, 51(1-2):35-47, 2012. Google Scholar
  7. Erik Palmgren. Constructions of categories of setoids from proof-irrelevant families. Arch. Math. Logic, 56(1-2):51-66, 2017. Google Scholar
  8. Erik Palmgren and Olov Wilander. Constructing categories and setoids of setoids in type theory. Log. Methods Comput. Sci., 10(3):3:25, 14, 2014. Google Scholar
  9. The Univalent Foundations Program. Homotopy type theory - univalent foundations of mathematics. The Univalent Foundations Program, Princeton, NJ; Institute for Advanced Study (IAS), Princeton, NJ, 2013. Google Scholar
  10. Olov Wilander. Constructing a small category of setoids. Math. Structures Comput. Sci., 22(1):103-121, 2012. Google Scholar
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