An equational variant of Lawvere's natural numbers object

A natural numbers object 1 (0) under right arrow N (S) under right arrow 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 S is unique o nly up to quasi-equality, where two arrows N --> A are called quasi-equal i f they are equalized by the canonical arrow A --> N-(NA). We show that quas i-natural numbers objects can be characterized equationally. (C) 2000 Elsev ier Science B.V. All rights reserved. MSC: 18C99; 18D99.