delooping of the additive monoid of natural numbers
- notation:
- objects: a single object
- morphisms: the natural numbers, with addition serving as composition
- Related categories: ,
Every monoid induces a one-object category with morphisms given by the elements of , and composition given by the monoid operation. Some of the properties of this category depend on the specific monoid. In this example, we take the commutative monoid .
Properties
Properties from the database
Deduced properties
- is essentially small
- is locally small
- is locally essentially small
- is well-copowered
- is well-powered
- is Cauchy complete
- is inhabited
- has pushouts
- has a cogenerator
- is right cancellative
Non-Properties
Non-Properties from the database
- does not have equalizers
- is not essentially finite
- does not have sequential limits
- does not have zero morphisms
Deduced Non-Properties*
- is not finite
- is not trivial
- is not essentially discrete
- is not discrete
- is not thin
- does not have products
- does not have coequalizers
- is not complete
- is not finitely complete
- does not have binary products
- does not have finite products
- does not have a terminal object
- is not pointed
- does not have connected limits
- does not have countable products
- does not have filtered limits
- does not have wide pullbacks
- does not have exact filtered colimits
- is not infinitary distributive
- is not distributive
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not finitary algebraic
- is not an elementary topos
- is not cartesian closed
- is not a Grothendieck topos
- does not have a subobject classifier
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not a groupoid
- is not balanced
- is not mono-regular
- is not Malcev
- does not have coproducts
- does not have disjoint coproducts
- is not cocomplete
- is not finitely cocomplete
- does not have binary coproducts
- does not have finite coproducts
- does not have disjoint finite coproducts
- does not have an initial object
- does not have a strict initial object
- does not have connected colimits
- does not have a strict terminal object
- does not have countable coproducts
- is not epi-regular
- does not have filtered colimits
- does not have wide pushouts
- does not have sequential colimits
*This also uses the deduced properties.
Unknown properties
—
Special morphisms
- Isomorphisms: only the number
- Monomorphisms: every morphism
- Epimorphisms: every morphism