category of countable sets

Satisfied properties

Assigned properties

• is locally small
• is essentially small
• is finitely complete
• is finitely cocomplete
• has a subobject classifier
• has an extremal generator
• has an extremal cogenerator
• is semi-strongly connected
• is countably extensive
• has effective congruences
• is regular
• is coregular

Deduced properties

• has equalizers
• has finite products
• is connected
• has countable coproducts
• is extensive
• is filtered
• has an extremal generating collection
• has a generator
• has coequalizers of kernel pairs
• has pullback-stable regular epimorphisms
• is Barr-exact
• is locally essentially small
• is well-powered
• is mono-regular
• has a regular subobject classifier
• has coequalizers
• has finite coproducts
• is cofiltered
• has a cogenerator
• has an extremal cogenerating collection
• has equalizers of cokernel pairs
• has pushout-stable regular monomorphisms
• is well-copowered
• has quotients of congruences
• is inhabited
• has coreflexive equalizers
• is Cauchy complete
• has disjoint finite coproducts
• has a strict initial object
• is distributive
• is countably distributive
• is sifted
• is ℵ₁-filtered
• has a generating collection
• is balanced
• has binary products
• has a terminal object
• has finite powers
• has kernel pairs
• has pullbacks
• is a pretopos
• has reflexive coequalizers
• is cosifted
• has sequential colimits
• has a cogenerating collection
• has binary coproducts
• has an initial object
• has countable copowers
• has finite copowers
• has cokernel pairs
• has pushouts
• has a natural numbers object
• has a parametrized natural numbers object
• is accessible
• has a multi-terminal object
• has effective cocongruences
• has binary powers
• is epi-regular
• is concretizable
• has a conatural numbers object
• is coaccessible
• has a multi-initial object
• has coquotients of cocongruences
• is ℵ₁-cofiltered
• has binary copowers
• is co-Malcev
• is Barr-coexact

Unsatisfied properties

Assigned properties

• is not small
• is not skeletal
• does not have countable powers
• does not have ℵ₂-small copowers
• does not have ℵ₁-cofiltered limits

Deduced properties*

• is not cartesian closed
• does not have countable products
• does not have ℵ₂-small powers
• does not have sequential limits
• is not gaunt
• is not direct
• is not ℵ₁-coaccessible
• does not have cofiltered limits
• does not have ℵ₂-small coproducts
• does not have copowers
• is not inverse
• is not an elementary cotopos
• is not self-dual
• is not locally cartesian closed
• is not trivial
• is not discrete
• does not have directed limits
• is not a groupoid
• does not have ℵ₂-small products
• does not have powers
• does not have filtered colimits
• does not have ℵ₁-filtered colimits
• does not have wide pullbacks
• is not an elementary topos
• is not a Grothendieck topos
• is not locally ℵ₁-copresentable
• is not finitely coaccessible
• is not a dual generalized variety
• is not countably codistributive
• does not have exact cofiltered limits
• does not have cocartesian cofiltered limits
• does not have cofiltered-limit-stable epimorphisms
• is not countably coextensive
• does not have cosifted limits
• does not have coproducts
• is not thin
• is not pointed
• is not ℵ₁-accessible
• is not finitely accessible
• is not locally poly-presentable
• is not Grothendieck abelian
• is not core-connected
• is not essentially discrete
• does not have disjoint coproducts
• does not have a strict terminal object
• is not infinitary distributive
• does not have exact filtered colimits
• does not have cartesian filtered colimits
• does not have filtered-colimit-stable monomorphisms
• does not satisfy CIP
• is not infinitary extensive
• is not left cancellative
• does not have directed colimits
• does not have sifted colimits
• does not have products
• does not have connected limits
• is not essentially finite
• is not regular-subobject-trivial
• is not retract-trivial
• is not core-thin
• is not a quasitopos
• is not locally finitely copresentable
• is not dually finitary algebraic
• is not locally finitely multi-copresentable
• is not co-Grothendieck abelian
• is not dually multi-algebraic
• is not cocartesian coclosed
• is not cocomplete
• does not have disjoint finite products
• is not infinitary codistributive
• does not satisfy CSP
• is not infinitary coextensive
• is not right cancellative
• does not have wide pushouts
• does not have a regular quotient object classifier
• is not regular-quotient-trivial
• is not locally finite
• is not essentially countable
• is not a Grothendieck cotopos
• is not unital
• is not locally finitely presentable
• is not locally ℵ₁-presentable
• is not locally presentable
• is not finitary algebraic
• is not locally multi-presentable
• is not locally finitely multi-presentable
• is not a generalized variety
• is not complete
• is not strongly connected
• does not have connected colimits
• does not have zero morphisms
• is not finite
• is not countable
• is not subobject-trivial
• is not Malcev
• is not one-way
• is not total
• is not counital
• is not locally copresentable
• is not locally poly-copresentable
• is not dually one-sorted finitary algebraic
• is not locally cocartesian coclosed
• is not multi-cocomplete
• does not have disjoint products
• is not codistributive
• is not coextensive
• does not have a quotient object classifier
• is not quotient-trivial
• is not a quasicotopos
• is not cototal
• is not additive
• does not have biproducts
• is not preadditive
• is not one-sorted finitary algebraic
• is not multi-algebraic
• is not multi-complete
• does not have kernels
• is not normal
• does not have a coparametrized conatural numbers object
• is not locally multi-copresentable
• does not have cokernels
• is not conormal
• is not a precotopos
• is not abelian
• is not split abelian

*This also uses the deduced satisfied properties.