delooping of the additive monoid of ordinal numbers
- notation:
- objects: a single object
- morphisms: ordinal numbers, with addition as composition
- Related categories:
Every monoid induces a one-object category . This also works when is large, in which case is not locally small. In this example, we apply this construction to the large monoid of ordinal numbers with respect to addition.
Properties
Properties from the database
- has a cogenerator
- is connected
- has equalizers
- has a generator
- is left cancellative
- is skeletal
- is well-copowered
Deduced properties
- is Cauchy complete
- is inhabited
Non-Properties
Non-Properties from the database
- is not balanced
- does not have binary coproducts
- does not have binary products
- does not have an initial object
- is not locally essentially small
- is not right cancellative
- does not have a terminal object
- is not well-powered
- does not have zero morphisms
Deduced Non-Properties*
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not locally small
- is not discrete
- is not essentially discrete
- is not trivial
- does not have finite products
- is not finitely complete
- is not complete
- does not have products
- is not pointed
- does not have a strict initial object
- does not have countable products
- 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 mono-regular
- is not Malcev
- is not thin
- does not have coequalizers
- is not cocomplete
- is not finitely cocomplete
- does not have finite coproducts
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- does not have coproducts
- does not have connected colimits
- does not have a strict terminal object
- does not have countable coproducts
- is not epi-regular
- is not self-dual
*This also uses the deduced properties.
Unknown properties
For these properties the database currently doesn't have an answer if they are satisfied or not. Please help to complete the data!
- has connected limits
- has filtered colimits
- has filtered limits
- has pullbacks
- has pushouts
- has sequential colimits
- has sequential limits
- has wide pullbacks
- has wide pushouts
Special morphisms
- Isomorphisms: only the ordinal
- Monomorphisms: every ordinal number
- Epimorphisms: finite ordinal numbers
Comments
- Lots of properties are unknown here. Please help to fill in the gaps!