Undecidability of Equality in the Free Locally Cartesian Closed Category

Abstract : We show that a version of Martin-Löf type theory with extensional identity, a unit type N 1 , Σ, Π, 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 then show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic.
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Simon Castellan, Pierre Clairambault, Peter Dybjer. Undecidability of Equality in the Free Locally Cartesian Closed Category. TLCA 2015 13th International Conference on Typed Lambda Calculi and Applications, Jul 2015, Varsovie, Poland. 13th International Conference on Typed Lambda Calculi and Applications, 〈http://rdp15.mimuw.edu.pl/index.php?site=tlca〉. 〈10.4230/LIPIcs.TLCA.2015.138〉. 〈hal-01286479〉

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