Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended Version)

Abstract : We show that a version of Martin-Löf type theory with an extensional identity type former I, a unit type N1, Σ-types, Π-types, and a base type is a free category with families (supporting these type formers) both in a 1-and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic. Essentially the same construction also shows a slightly strengthened form of the result that equality in extensional Martin-Löf type theory with one universe is undecidable.
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Submitted on : Thursday, August 31, 2017 - 10:23:48 AM
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Simon Castellan, Pierre Clairambault, Peter Dybjer. Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended Version). 2017. ⟨hal-01579415⟩

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