The groupoid interpretation of type theory

Abstract

Many will agree that identity sets are the most intriguing concept of intensional Martin-Löf type theory. For instance, it may appear surprising that their axiomatisation as an inductive family allows one to deduce the usual properties of equality, notably the replacement rule (Leibniz’s principle) which gives P(a′) from P(a) and a proof that a equals a′. This holds for arbitrary families of sets P, not only those corresponding to a predicate. This is not in conflict with decidability of type checking since if a equals a′ and p : P(a) then one does not in general have p : P(a′), but only subst(s, p) : P(a′) where s is the proof that a equals a′ and subst is defined from the eliminator for identity sets. It is a natural question to ask whether these translation functions subst(s, _) actually depend upon the nature of the proof s or, more generally, the question whether any two elements of an identity set are equal. We will call UIP(A) (t/niqueness of Identity Proofs) the following property. If a1, a2 are objects of type A then for any two proofs p and q of the proposition “a1 equals a2” we can prove that p and q are equal. More generally, UIP will stand for UIP(A) for all types A. Note that in traditional logical formalism a principle like UIP cannot even be expressed sensibly as proofs cannot be referred to by terms of the object language and thus are not within the scope of prepositional equality. The question of whether UIP is valid in intensional Martin-Löf type theory was open for a while, though it was commonly believed that UIP is underivable as any attempt for constructing a proof has failed (Coquand 1992; Streicher 1993; Altenkirch 1992). On the other hand, the intuition that a type is determined by its canonical objects might be seen as evidence for the validity of UIP as the identity sets have at most one canonical element corresponding to an instance of reflexivity.