preordered set of integers w.r.t. divisibility

Satisfied Properties

Assigned properties

• is small
• is countable
• is thin
• is distributive
• is locally ℵ₁-presentable

Deduced properties

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

Unsatisfied Properties

Assigned properties

• is not skeletal
• is not self-dual
• is not countably distributive
• is not countably codistributive
• does not have an extremal generator
• does not have an extremal cogenerator

Deduced properties*

• is not core-connected
• is not trivial
• is not discrete
• is not infinitary distributive
• is not cartesian closed
• does not have cartesian filtered colimits
• is not countably extensive
• is not balanced
• is not strongly connected
• is not one-sorted finitary algebraic
• is not a groupoid
• is not gaunt
• is not direct
• is not infinitary codistributive
• is not cocartesian coclosed
• does not have cocartesian cofiltered limits
• is not countably coextensive
• is not inverse
• is not pointed
• is not locally cartesian closed
• does not have zero morphisms
• is not essentially discrete
• does not have disjoint finite coproducts
• does not have exact filtered colimits
• does not have biproducts
• is not extensive
• is not infinitary extensive
• is not mono-regular
• is not additive
• is not an elementary topos
• is not locally cocartesian coclosed
• does not have disjoint finite products
• does not have exact cofiltered limits
• is not coextensive
• is not infinitary coextensive
• is not epi-regular
• is not unital
• is not locally finitely presentable
• is not preadditive
• is not abelian
• is not Grothendieck abelian
• is not semi-strongly connected
• 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 a pretopos
• is not a quasitopos
• is not counital
• 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 finitely accessible
• is not finitary algebraic
• is not split abelian
• is not essentially finite
• is not a generalized variety
• is not locally finitely multi-presentable
• is not multi-algebraic
• is not finite

*This also uses the deduced satisfied properties.