We show that a version of Martin-Lof 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.
['Simon Castellan', 'Pierre Clairambault', 'Peter Dybjer']
Undecidability of Equality in the Free Locally Cartesian Closed Category