Logical Methods in Computer Science (Sep 2021)

W-types in setoids

  • Jacopo Emmenegger

DOI
https://doi.org/10.46298/lmcs-17(3:28)2021
Journal volume & issue
Vol. Volume 17, Issue 3

Abstract

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We present a construction of W-types in the setoid model of extensional Martin-L\"of type theory using dependent W-types in the underlying intensional theory. More precisely, we prove that the internal category of setoids has initial algebras for polynomial endofunctors. In particular, we characterise the setoid of algebra morphisms from the initial algebra to a given algebra as a setoid on a dependent W-type. We conclude by discussing the case of free setoids. We work in a fully intensional theory and, in fact, we assume identity types only when discussing free setoids. By using dependent W-types we can also avoid elimination into a type universe. The results have been verified in Coq and a formalisation is available on the author's GitHub page.

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