partially ordered set of natural numbers

Satisfied properties

Assigned properties

• is small
• is countable
• is thin
• is semi-strongly connected
• is direct

Deduced properties

• is locally cartesian closed
• is connected
• has cosifted limits
• is essentially small
• is locally small
• is essentially countable
• has equalizers
• has a generating collection
• is left cancellative
• is locally finite
• is one-way
• is skeletal
• has binary powers
• has binary products
• is locally cocartesian coclosed
• has coequalizers
• has a cogenerating collection
• is right cancellative
• has binary copowers
• has binary coproducts
• is a dual generalized variety
• is finitely coaccessible
• is Cauchy complete
• has kernel pairs
• has pullbacks
• has pullback-stable regular epimorphisms
• is inhabited
• has coreflexive equalizers
• is filtered
• has an extremal generating collection
• is locally essentially small
• is well-powered
• is regular-subobject-trivial
• is retract-trivial
• is core-thin
• has cokernel pairs
• has pushouts
• has pushout-stable regular monomorphisms
• has reflexive coequalizers
• is cofiltered
• has cofiltered limits
• has an extremal cogenerating collection
• is well-copowered
• is regular-quotient-trivial
• is accessible
• is ℵ₁-accessible
• has quotients of congruences
• has effective congruences
• is sifted
• has wide pullbacks
• is concretizable
• has a generator
• is gaunt
• has cofiltered-limit-stable epimorphisms
• is ℵ₁-coaccessible
• is coaccessible
• has effective cocongruences
• has coquotients of cocongruences
• is cosifted
• has directed limits
• has ℵ₁-cofiltered limits
• has a cogenerator
• has ℵ₁-filtered colimits
• is locally poly-presentable
• has coequalizers of kernel pairs
• has sequential limits
• has connected limits
• has equalizers of cokernel pairs
• is locally multi-presentable
• is multi-cocomplete
• has a multi-initial object
• has an initial object
• has a strict initial object
• is cocartesian coclosed
• is ℵ₁-cofiltered
• has finite coproducts
• has copowers
• has a conatural numbers object
• has a coparametrized conatural numbers object
• has cocartesian cofiltered limits
• is finitely cocomplete
• has finite copowers
• has ℵ₂-small copowers
• is co-Malcev
• has exact cofiltered limits
• has countable copowers
• is coregular
• has a regular quotient object classifier
• is Barr-coexact

Unsatisfied properties

Assigned properties

• does not have countable coproducts
• does not have an extremal generator
• does not have an extremal cogenerator

Deduced properties*

• is not core-connected
• is not countably distributive
• is not countably extensive
• is not balanced
• is not one-sorted finitary algebraic
• is not dually one-sorted finitary algebraic
• does not have ℵ₂-small coproducts
• does not have sequential colimits
• is not trivial
• is not infinitary distributive
• is not infinitary extensive
• is not mono-regular
• is not an elementary topos
• does not have directed colimits
• is not epi-regular
• does not have coproducts
• is not self-dual
• is not Grothendieck abelian
• is not extensive
• is not strongly connected
• does not have disjoint coproducts
• does not satisfy CIP
• does not have filtered colimits
• is not a groupoid
• is not pointed
• is not normal
• does not have a subobject classifier
• is not subobject-trivial
• does not have disjoint finite coproducts
• is not cocomplete
• is not coextensive
• does not satisfy CSP
• is not infinitary coextensive
• is not conormal
• does not have a quotient object classifier
• is not quotient-trivial
• does not have disjoint finite products
• is not essentially finite
• is not unital
• is not locally finitely presentable
• is not locally ℵ₁-presentable
• is not locally presentable
• is not finitely accessible
• is not finitary algebraic
• is not abelian
• is not additive
• does not have zero morphisms
• is not essentially discrete
• does not have exact filtered colimits
• does not have cartesian filtered colimits
• does not have filtered-colimit-stable monomorphisms
• does not have sifted colimits
• is not finite
• is not complete
• is not a Grothendieck topos
• is not a pretopos
• is not total
• is not counital
• is not locally copresentable
• does not have a terminal object
• does not have disjoint products
• is not countably coextensive
• does not have wide pushouts
• is not an elementary cotopos
• is not a Grothendieck cotopos
• is not a precotopos
• is not cototal
• does not have a natural numbers object
• is not locally finitely multi-presentable
• is not preadditive
• is not split abelian
• does not have biproducts
• is not a generalized variety
• does not have products
• does not have a multi-terminal object
• is not multi-complete
• is not discrete
• does not have kernels
• is not ℵ₁-filtered
• does not have connected colimits
• does not have finite products
• does not have finite powers
• is not locally finitely copresentable
• is not locally ℵ₁-copresentable
• is not co-Grothendieck abelian
• is not dually finitary algebraic
• is not locally poly-copresentable
• is not inverse
• does not have cokernels
• does not have a strict terminal object
• does not have a parametrized natural numbers object
• is not multi-algebraic
• is not cartesian closed
• is not finitely complete
• is not distributive
• does not have countable products
• does not have countable powers
• is not locally multi-copresentable
• is not locally finitely multi-copresentable
• is not dually multi-algebraic
• is not infinitary codistributive
• is not codistributive
• is not Malcev
• does not have ℵ₂-small products
• does not have ℵ₂-small powers
• is not regular
• does not have a regular subobject classifier
• is not a quasitopos
• is not countably codistributive
• is not a quasicotopos
• does not have powers
• is not Barr-exact

*This also uses the deduced satisfied properties.