poset of ordinal numbers
This is a large variant of the poset of natural numbers.
Satisfied Properties
Assigned properties
- is thin
- is locally small
- is cocomplete
- has binary products
- has connected limits
- is well-powered
- is semi-strongly connected
- is direct
Deduced properties
- is locally cartesian closed
- is connected
- has sequential limits
- has binary powers
- has equalizers
- has wide pullbacks
- is locally essentially small
- has a generating set
- is left cancellative
- is locally finite
- is one-way
- is skeletal
- is locally cocartesian coclosed
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is multi-cocomplete
- has cosifted limits
- has a cogenerating set
- is right cancellative
- has binary copowers
- is Cauchy complete
- has coreflexive equalizers
- has effective cocongruences
- has effective congruences
- has reflexive coequalizers
- has pullbacks
- is inhabited
- is filtered
- has sifted colimits
- has cofiltered limits
- is core-thin
- is co-Malcev
- has pushouts
- is coregular
- has finite coproducts
- has a multi-initial object
- is cofiltered
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- has a regular quotient object classifier
- has quotients of congruences
- is sifted
- has filtered colimits
- has a generator
- is gaunt
- has an initial object
- has coquotients of cocongruences
- is Barr-coexact
- has cofiltered-limit-stable epimorphisms
- is cosifted
- has directed limits
- has ℵ₁-cofiltered limits
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has finite copowers
- has a cogenerator
- has filtered-colimit-stable monomorphisms
- is ℵ₁-filtered
- has directed colimits
- has ℵ₁-filtered colimits
- has a strict initial object
- is cocartesian coclosed
- is ℵ₁-cofiltered
- has sequential colimits
- has countable copowers
- has cocartesian cofiltered limits
- has exact cofiltered limits
Unsatisfied Properties
Assigned properties
- does not have a terminal object
- is not well-copowered
- is not inverse
Deduced properties*
- is not accessible
- is not finitary algebraic
- does not have a multi-terminal object
- is not essentially finite
- is not essentially countable
- does not have finite products
- does not have finite powers
- is not essentially small
- is not an elementary topos
- is not coaccessible
- is not discrete
- is not a groupoid
- is not pointed
- does not have a strict terminal object
- does not have a quotient object classifier
- is not finite
- is not self-dual
- is not unital
- does not have a natural numbers object
- is not locally presentable
- is not ℵ₁-accessible
- is not locally multi-presentable
- is not locally poly-presentable
- is not preadditive
- is not additive
- does not have biproducts
- is not cartesian closed
- is not finitely complete
- is not multi-complete
- is not trivial
- is not essentially discrete
- is not infinitary distributive
- is not countably distributive
- is not distributive
- does not have cartesian filtered colimits
- is not balanced
- does not have zero morphisms
- does not have countable products
- does not have countable powers
- is not small
- is not countable
- is not a Grothendieck topos
- is not counital
- is not locally copresentable
- does not have disjoint finite products
- is not codistributive
- is not coextensive
- is not epi-regular
- is not Malcev
- is not locally ℵ₁-presentable
- is not Grothendieck abelian
- is not finitely accessible
- is not abelian
- is not a generalized variety
- is not complete
- is not regular
- is not strongly connected
- does not have disjoint finite coproducts
- does not have kernels
- does not have exact filtered colimits
- does not satisfy CIP
- is not mono-regular
- is not normal
- does not have ℵ₂-small products
- does not have ℵ₂-small powers
- does not have a subobject classifier
- does not have a regular subobject classifier
- does not have disjoint products
- is not countably codistributive
- does not have cokernels
- does not satisfy CSP
- is not infinitary coextensive
- is not conormal
- is not quotient-trivial
- is not locally finitely presentable
- is not locally strongly finitely presentable
- is not locally finitely multi-presentable
- is not split abelian
- is not multi-algebraic
- does not have products
- is not extensive
- is not Barr-exact
- does not have disjoint coproducts
- does not have powers
- is not subobject-trivial
- is not infinitary codistributive
- is not infinitary extensive
- is not a pretopos
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- initial object:
- coproducts: supremum
Special morphisms
- isomorphisms: only the identities
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as isomorphisms
- regular epimorphisms: same as isomorphisms