pointed
A category is pointed when it has a zero object, i.e. an object which is both initial and terminal.
- Dual property: pointed (self-dual)
- Related properties: initial object, terminal object
- nLab Link
Relevant implications
pointed is equivalent to zero morphisms and initial object
easypointed and cartesian closed implies trivial
We have for every object .strict initial object and pointed implies trivial
If is the zero object, then for every object the unique morphism is an isomorphism by assumption.pointed is equivalent to zero morphisms and terminal object
[dualized] easystrict terminal object and pointed implies trivial
[dualized] If is the zero object, then for every object the unique morphism is an isomorphism by assumption.
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of finite abelian groups
- category of finitely generated abelian groups
- category of free abelian groups
- category of groups
- category of left R-modules
- category of monoids
- category of pointed sets
- category of rngs
- category of sets and relations
- category of vector spaces
- trivial category
- walking isomorphism
Counterexamples
- category of combinatorial species
- category of commutative rings
- category of fields
- category of finite orders
- category of finite sets
- category of finite sets and bijections
- category of finite sets and injections
- category of finite sets and surjections
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with ∞ allowed
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of non-empty sets
- category of posets
- category of rings
- category of schemes
- category of sets
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category on two objects
- empty category
- partial order [0,1]
- partial order of extended natural numbers
- partial order of natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- walking morphism
- walking parallel pair of morphisms
Unknown
For these categories the database has no info if they satisfy this property or not.
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