natural numbers object
A natural numbers object (NNO) in a category with finite products is a triple
satisfying the following universal property: for all , there is a unique such that and in element notation.
This concept is an abstraction of the set of natural numbers, which indeed provide a NNO for the category of sets. We have used the parametrized definition here which 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
- nLab Link
Relevant implications
- countably distributive implies natural numbers object
- finite products andthin implies natural numbers object
- natural numbers object andpointed implies trivial
- natural numbers object andstrict terminal object implies one-way
- natural numbers object implies finite products
Examples
There are 26 categories with this property.
- category of Hausdorff spaces
- 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 posets
- category of prosets
- category of schemes
- category of sets
- category of sheaves
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of Z-functors
- poset [0,1]
- poset of extended natural numbers
- proset of integers w.r.t. divisibility
- trivial category
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism
Counterexamples
There are 44 categories without this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of fields
- 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 finitely generated abelian groups
- category of free abelian groups
- category of groups
- category of left modules over a division 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 sets and relations
- category of vector spaces
- delooping of a non-trivial finite group
- delooping of an infinite countable 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
- empty category
- poset of natural numbers
- poset of ordinal numbers
- simplex category
- 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.
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