An equational variant of Lawvere's natural numbers object

Abstract

A natural numbers object 1 → 0 N → S N in a cartesian closed category associates to each pair of arrows a:1→ A and h: A→ A a unique arrow f: N→ A such that f0= a and fS= hf. We call ( N,0, S) a quasi-natural numbers object if the arrow f is unique only up to quasi-equality, where two arrows N→ A are called quasi-equal if they are equalized by the canonical arrow A→ N ( N A ) . We show that quasi-natural numbers objects can be characterized equationally.