CatDat

parametrized natural numbers object

A parametrized natural numbers object in a category with finite products is a triple $$(N,\, z : 1 \to N,\, s : N \to N)$$ satisfying the following universal property: for all $f : A \to X$, $g : X \to X$ there is a unique $\Phi : A \times N \to X$ such that $\Phi(a,z)=f(a)$ and $\Phi(a,s(n)) = g(\Phi(a,n))$ in element notation.
This concept is an abstraction of the set of natural numbers, which indeed provides a parametrized natural numbers object for the category of sets. If we specialize the universal property to $A=1$, we get the notion of a natural numbers object. The parametrized version is more natural (sic!) for categories that are not cartesian closed (cf. Johnstone, Part A, Remark 2.5.3).

• Related properties: elementary topos, finite products, natural numbers object
• nLab Link

Relevant implications

• cartesian closed andnatural numbers object implies parametrized natural numbers object
• countably distributive implies parametrized natural numbers object
• finite products andone-way implies parametrized natural numbers object
• parametrized natural numbers object andpointed implies trivial
• parametrized natural numbers object andstrict terminal object implies one-way
• parametrized natural numbers object implies finite products
• parametrized natural numbers object implies natural numbers object

Examples

There are 30 categories with this property.

• category of countable sets
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with ∞ allowed
• category of non-empty sets
• category of pairs of sets
• category of partially ordered sets
• category of preordered sets
• category of schemes
• category of set functions and commutative squares
• category of sets
• 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 Z-functors
• 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

Counterexamples

There are 67 categories without 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 combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• category of fields
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• 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-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 free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• 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 metric spaces with non-expansive maps
• category of monoids
• category of pointed sets
• category of pointed topological spaces
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of semigroups
• category of sequences of abelian groups
• category of sets and relations
• category of sets with finite-to-one maps
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• 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 on two objects
• dual of the category of sets
• dual of the category of topological spaces
• 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
• walking splitting

Unknown

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

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