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 category with a single object . 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, so composition is .
Satisfied Properties
Properties from the database
- has cofiltered limits
- has a cogenerator
- has equalizers
- has a generator
- is left cancellative
- is locally cartesian closed
- has pullbacks
- is skeletal
- is strongly connected
- is well-copowered
Deduced properties
- has coreflexive equalizers
- is Cauchy complete
- has wide pullbacks
- has connected limits
- has a generating set
- is inhabited
- has reflexive coequalizers
- is connected
- has cosifted limits
- has directed limits
- has sequential limits
- has a cogenerating set
Unsatisfied Properties
Properties from the database
- is not balanced
- does not have binary powers
- does not have an initial object
- is not locally essentially small
- is not one-way
- does not have pushouts
- is not right cancellative
- does not have sequential colimits
- does not have a terminal object
- is not well-powered
Deduced properties*
- does not have finite products
- does not have products
- does not have countable products
- is not complete
- is not finitely complete
- does not have binary products
- does not have finite powers
- does not have countable powers
- does not have powers
- does not have biproducts
- is not pointed
- does not have exact filtered colimits
- is not infinitary distributive
- is not distributive
- does not have a strict initial object
- is not extensive
- is not infinitary extensive
- is not regular
- is not lextensive
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not locally small
- is not a groupoid
- is not mono-regular
- is not direct
- is not thin
- does not have coequalizers
- does not have zero morphisms
- does not have binary copowers
- is not discrete
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not essentially discrete
- is not trivial
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not cartesian closed
- is not an elementary topos
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not a Grothendieck topos
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- is not Malcev
- is not unital
- does not have binary coproducts
- does not have finite coproducts
- does not have coproducts
- does not have countable coproducts
- does not have directed colimits
- does not have filtered colimits
- does not have sifted colimits
- does not have connected colimits
- is not cocomplete
- is not finitely cocomplete
- does not have wide pushouts
- does not have finite copowers
- does not have countable copowers
- does not have copowers
- does not have disjoint products
- does not have disjoint finite products
- is not infinitary codistributive
- is not codistributive
- does not have a strict terminal object
- is not coextensive
- is not infinitary coextensive
- is not coregular
- is not epi-regular
- is not inverse
- is not co-Malcev
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
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Special morphisms
- isomorphisms: only the ordinal
- monomorphisms: every ordinal number
- epimorphisms: finite ordinal numbers
- regular monomorphisms: ordinals of the form , where is any ordinal
- regular epimorphisms: same as isomorphisms