CatDat

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

  easy
• pointed and  cartesian closed implies   trivial

  We have $X \cong X \times 1 \cong X \times 0 \cong 0$ for every object $X$.
• strict initial object and  pointed implies   trivial

  If $0$ is the zero object, then for every object $A$ the unique morphism $A \to 0$ is an isomorphism by assumption.
• pointed is equivalent to   zero morphisms and  terminal object

  [dualized] easy
• strict terminal object and  pointed implies   trivial

  [dualized] If $0$ is the zero object, then for every object $A$ the unique morphism $A \to 0$ 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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