natural numbers object

A natural numbers object in a category with a terminal object $1$ is a triple $$(N,\, z : 1 \to N,\, s : N \to N)$$ satisfying the following universal property: for all $a : 1 \to X$, $g : X \to X$ there is a unique $\Phi : N \to X$ such that $\Phi \circ z = a$ and $\Phi \circ s = g \circ \Phi$. In element notation, this can also be written as $\Phi(z)=a$ and $\Phi(s(n)) = g(\Phi(n))$.
This concept is an abstraction of the set of natural numbers, which indeed provides a natural numbers object for the category of sets. There is also a parametrized version, which is more useful in categories that are not cartesian closed. Also, the results below show that "too many" categories have a natural numbers object.

Relevant implications

• cartesian closed andnatural numbers object implies parametrized natural numbers object
• countable copowers andterminal object implies natural numbers object
• natural numbers object implies terminal object
• one-way andterminal object implies natural numbers object
• parametrized natural numbers object implies natural numbers object
• pointed implies natural numbers object
• strict terminal object implies natural numbers object

Examples

There are 79 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of coproducts of Euclidean spaces
• category of countable groups
• category of countable sets
• category of countable-dimensional vector spaces
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite-dimensional vector spaces
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of finitely generated free abelian groups
• category of finitely generated free modules over Z x Z
• category of finitely generated projective modules over the ring of dual numbers
• category of free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of large vector spaces over a large field with a small basis
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of locally ringed spaces
• category of long transfinite sequences of abelian groups
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of partially ordered sets
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of set-indexed families of abelian groups
• category of sets
• category of sets and relations
• category of sets with a distinguished subset
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• category of Z-functors
• cocompletion of a discrete–pair join
• dual of the category of sets
• dual of the category of topological spaces
• partially ordered set of extended natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism
• walking splitting

Counterexamples

There are 33 categories without this property.

• category of combinatorial species
• category of fields
• category of fields of characteristic zero
• category of finite ordered sets
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd cardinality
• category of metric spaces with non-expansive maps
• category of partially ordered sets without isolated points
• category of pseudo-metric spaces with non-expansive maps
• category of sets with finite-to-one maps
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category of sets
• discrete category on two objects
• empty category
• forked commutative square
• partially ordered collection of ordinal numbers
• partially ordered set of natural numbers
• simplex category
• walking coreflexive pair
• walking fork
• walking idempotent
• walking parallel pair
• walking span

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

—