category of pointed sets
- notation:
- objects: pointed sets
- morphisms: pointed maps
- nLab Link
- Related categories:
This is the category of sets with a distinguished element, often called the base point. A map is called pointed when it preserves the base point.
Properties
Properties from the database
- has a cogenerator
- has disjoint coproducts
- is epi-regular
- is finitary algebraic
- is locally small
- is pointed
- has a subobject classifier
Deduced properties
- is locally essentially small
- has an initial object
- has zero morphisms
- has coproducts
- has disjoint finite coproducts
- has finite coproducts
- is locally finitely presentable
- is locally presentable
- is cocomplete
- is complete
- has connected limits
- has filtered limits
- is finitely complete
- has wide pullbacks
- has equalizers
- has products
- has finite products
- has countable products
- has binary products
- has a terminal object
- has pullbacks
- is connected
- is Cauchy complete
- has sequential limits
- has a generator
- is well-copowered
- is well-powered
- has exact filtered colimits
- has filtered colimits
- is locally ℵ₁-presentable
- is mono-regular
- is inhabited
- is balanced
- has connected colimits
- is finitely cocomplete
- has wide pushouts
- has coequalizers
- has countable coproducts
- has binary coproducts
- has pushouts
- has sequential colimits
Non-Properties
Non-Properties from the database
Deduced Non-Properties*
- is not discrete
- is not additive
- is not preadditive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not trivial
- is not essentially discrete
- is not cartesian closed
- is not thin
- is not essentially small
- is not small
- is not essentially finite
- is not finite
- is not left cancellative
- does not have a strict initial object
- is not right cancellative
- is not distributive
- is not infinitary distributive
- is not self-dual
- is not an elementary topos
- is not a Grothendieck topos
- is not a groupoid
- does not have a strict terminal object
*This also uses the deduced properties.
Unknown properties
—
Special morphisms
- Isomorphisms: bijective pointed maps
- Monomorphisms: injective pointed maps
- Epimorphisms: surjective pointed maps