partially ordered set of extended natural numbers
This category is a completed version of the thin category of natural numbers.
Satisfied Properties
Assigned properties
- is small
- is countable
- has coproducts
- is semi-strongly connected
- is direct
- is finitary algebraic
Deduced properties
- is locally finitely presentable
- is cocomplete
- is a generalized variety
- is regular
- is multi-algebraic
- is connected
- has sequential limits
- is essentially small
- is locally small
- is essentially countable
- is one-way
- is skeletal
- has copowers
- has ℵ₂-small coproducts
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has sifted colimits
- is ℵ₁-accessible
- has filtered-colimit-stable monomorphisms
- is locally finitely multi-presentable
- is multi-cocomplete
- has effective congruences
- has coequalizers of kernel pairs
- is finitely complete
- is inhabited
- has reflexive coequalizers
- has an extremal generating collection
- is locally essentially small
- is well-copowered
- is well-powered
- is Cauchy complete
- is core-thin
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coreflexive equalizers
- has countable coproducts
- has ℵ₂-small copowers
- has an extremal cogenerating collection
- is thin
- is Malcev
- is locally presentable
- is accessible
- has ℵ₁-filtered colimits
- has filtered colimits
- has connected limits
- is locally cartesian closed
- has equalizers
- has finite products
- has quotients of congruences
- is Barr-exact
- has cartesian filtered colimits
- is filtered
- is ℵ₁-filtered
- has a generating collection
- has kernel pairs
- is regular-subobject-trivial
- is left cancellative
- is locally finite
- has a generator
- is gaunt
- has binary powers
- is co-Malcev
- is coaccessible
- is locally cocartesian coclosed
- has finite coproducts
- has a multi-initial object
- has coquotients of cocongruences
- has effective cocongruences
- is cofiltered
- has sequential colimits
- has a cogenerating collection
- has countable copowers
- has wide pushouts
- is regular-quotient-trivial
- is right cancellative
- has a cogenerator
- is complete
- is coregular
- has binary copowers
- has a parametrized natural numbers object
- is locally multi-presentable
- has pullbacks
- has products
- is multi-complete
- is sifted
- has a terminal object
- has directed colimits
- has binary products
- has countable products
- has finite powers
- has wide pullbacks
- is concretizable
- has a regular subobject classifier
- is total
- is locally copresentable
- has pushouts
- has an initial object
- has equalizers of cokernel pairs
- is Barr-coexact
- is cosifted
- has cosifted limits
- has binary coproducts
- has finite copowers
- has cokernel pairs
- has a regular quotient object classifier
- is cototal
- has a natural numbers object
- is locally poly-presentable
- has a strict initial object
- is cartesian closed
- has a multi-terminal object
- has powers
- has ℵ₂-small products
- has countable powers
- has cofiltered limits
- is a quasitopos
- has a strict terminal object
- is cocartesian coclosed
- is ℵ₁-cofiltered
- is distributive
- is countably distributive
- is infinitary distributive
- has ℵ₂-small powers
- has cocartesian cofiltered limits
- is codistributive
- is countably codistributive
- is infinitary codistributive
- has cofiltered-limit-stable epimorphisms
- has directed limits
- has ℵ₁-cofiltered limits
- has exact cofiltered limits
Unsatisfied Properties
Assigned properties
- is not essentially finite
- is not self-dual
- is not inverse
- does not have an extremal generator
- does not have an extremal cogenerator
Deduced properties*
- is not core-connected
- is not trivial
- is not balanced
- is not strongly connected
- is not one-sorted finitary algebraic
- is not a groupoid
- is not finite
- is not discrete
- is not pointed
- does not have zero morphisms
- is not essentially discrete
- does not have disjoint finite coproducts
- is not mono-regular
- is not additive
- is not an elementary topos
- does not have disjoint finite products
- is not epi-regular
- is not unital
- is not preadditive
- is not abelian
- does not have biproducts
- is not extensive
- does not have disjoint coproducts
- does not have kernels
- does not satisfy CIP
- is not normal
- does not have a subobject classifier
- is not subobject-trivial
- is not a Grothendieck topos
- is not counital
- is not coextensive
- does not have disjoint products
- does not have cokernels
- does not satisfy CSP
- is not conormal
- does not have a quotient object classifier
- is not quotient-trivial
- is not Grothendieck abelian
- is not split abelian
- is not countably extensive
- is not a pretopos
- is not countably coextensive
- is not infinitary extensive
- is not infinitary coextensive
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
- terminal object:
- initial object:
- products: infimum
- coproducts: supremum
Special morphisms
- isomorphisms: only the identity morphisms
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as isomorphisms
- regular epimorphisms: same as isomorphisms
Functors
The database stores 1 functor based on the partially ordered set of extended natural numbers.