An equational variant of Lawvere's natural numbers object
type: Item
instance of: scholarly article
title: An equational variant of Lawvere's natural numbers object
zbMATH Open document ID: 0963.18004
DOI: 10.1016/S0022-4049(99)00178-4
published in: Journal of Pure and Applied Algebra
publication date: 2001-06-17T00:00:00Z
review text: As is well known, \textit{F. W. Lawvere}'s definition of a natural numbers object \(N\) in a cartesian closed category is equivalent to asserting that morphisms defined recursively over \(N\) exist and are unique. If one attempts to characterize recursion equationally, one can capture the existence of recursively defined morphisms but not their uniqueness (the latter requires an implication), leading to a notion that has been called a weak natural numbers object. In this paper, the author observes that one can recapture part of the uniqueness by exploiting the fact that addition and truncated subtraction enable one to define a Mal'cev operation on a natural numbers object; this leads to a notion which he calls a quasi-NNO. Morphisms defined recursively on a quasi-NNO are not unique in general, but they are unique if their codomain belongs to the class of objects which `admit enough \(N\)-valued functions to separate points' (in the internal logic of the cartesian closed category).
Mathematics Subject Classification ID: 18C99
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 1538810
zbMATH Keywords: natural numbers object in cartesian closed category
zbMATH Keywords: recursively defined morphisms
zbMATH Keywords: weak natural numbers object
zbMATH Keywords: Mal'cev operation
zbMATH Keywords: internal logic of cartesian closed category
reviewed by: Peter Johnstone
MaRDI profile type: Publication
cites work: ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES
cites work: Q5812175
cites work: Least fixpoints of endofunctors of cartesian closed categories
cites work: Q3727946
cites work: AN ELEMENTARY THEORY OF THE CATEGORY OF SETS
cites work: Adjointness in Foundations
cites work: Pre-recursive categories
author: J. Lambek
description: scientific article; zbMATH DE number 1538810