Undecidability of Equality in the Free Locally Cartesian Closed Category
Simon Castellan, Pierre Clairambault, Peter Dybjer · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2015
We show that a version of Martin-Löf type theory with extensional identity, a unit type N1, Sigma, Pi, 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.