category of non-empty sets

Satisfied properties

Assigned properties

• is locally small
• has an extremal generator
• has an extremal cogenerator
• has products
• is cartesian closed
• has kernel pairs
• has binary coproducts
• has equalizers of cokernel pairs
• is mono-regular
• is epi-regular
• has pushout-stable regular monomorphisms
• is strongly connected
• is finitely accessible
• is a generalized variety
• has a parametrized natural numbers object
• is multi-complete

Deduced properties

• has finite products
• has a natural numbers object
• has filtered-colimit-stable monomorphisms
• is ℵ₁-accessible
• has filtered colimits
• has sifted colimits
• has effective congruences
• has a multi-terminal object
• is semi-strongly connected
• is Cauchy complete
• has an extremal generating collection
• has a generator
• is balanced
• has powers
• has ℵ₂-small products
• is locally essentially small
• has a cogenerator
• has an extremal cogenerating collection
• has binary copowers
• has cokernel pairs
• has pushouts
• has a terminal object
• is accessible
• has ℵ₁-filtered colimits
• has cartesian filtered colimits
• is connected
• has disjoint finite products
• has directed colimits
• has reflexive coequalizers
• has a generating collection
• is inhabited
• has countable products
• has ℵ₂-small powers
• has binary products
• has finite powers
• has coequalizers
• has a cogenerating collection
• has wide pushouts
• is well-powered
• is well-copowered
• has quotients of congruences
• is sifted
• is filtered
• is ℵ₁-filtered
• has countable powers
• has binary powers
• is concretizable
• has disjoint products
• is cosifted
• has sequential colimits
• has connected colimits
• has coequalizers of kernel pairs

Unsatisfied properties

Assigned properties

• is not skeletal
• is not cofiltered
• does not have sequential limits
• does not have ℵ₁-cofiltered limits
• does not have coquotients of cocongruences
• is not coaccessible

Deduced properties*

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

*This also uses the deduced satisfied properties.