walking splitting

Satisfied properties

Assigned properties

• is small
• is finite
• is gaunt
• is self-dual
• is pointed
• has equalizers
• is normal
• has an extremal generator
• is preadditive
• has sifted colimits
• is a generalized variety

Deduced properties

• has a natural numbers object
• has zero morphisms
• is ℵ₁-accessible
• has filtered-colimit-stable monomorphisms
• has effective congruences
• has coreflexive equalizers
• is Cauchy complete
• has filtered colimits
• has reflexive coequalizers
• has an extremal generating collection
• has a generator
• has an initial object
• is mono-regular
• is essentially small
• is locally small
• is countable
• is essentially finite
• is core-thin
• is skeletal
• has a conatural numbers object
• has a terminal object
• has coequalizers
• has an extremal cogenerator
• is a dual generalized variety
• is conormal
• has cosifted limits
• is finitely accessible
• is accessible
• has ℵ₁-filtered colimits
• has a multi-terminal object
• has quotients of congruences
• has kernels
• is ℵ₁-filtered
• has directed colimits
• has a generating collection
• is inhabited
• is balanced
• is locally essentially small
• is well-powered
• is essentially countable
• is locally finite
• is finitely coaccessible
• is coaccessible
• is ℵ₁-coaccessible
• has cofiltered-limit-stable epimorphisms
• has effective cocongruences
• has a multi-initial object
• has coquotients of cocongruences
• has cokernels
• is ℵ₁-cofiltered
• has cofiltered limits
• has a cogenerator
• has an extremal cogenerating collection
• is epi-regular
• is well-copowered
• is strongly connected
• is filtered
• is concretizable
• has ℵ₁-cofiltered limits
• is cofiltered
• has directed limits
• has sequential colimits
• has a cogenerating collection
• is semi-strongly connected
• is sifted
• has sequential limits
• is cosifted
• is connected

Unsatisfied properties

Assigned properties

• is not retract-trivial

Deduced properties*

• is not regular-subobject-trivial
• is not core-connected
• is not one-way
• is not regular-quotient-trivial
• is not trivial
• is not subobject-trivial
• is not right cancellative
• is not thin
• is not direct
• is not quotient-trivial
• is not left cancellative
• is not inverse
• does not have a parametrized natural numbers object
• is not discrete
• is not essentially discrete
• is not infinitary extensive
• is not countably extensive
• is not a groupoid
• does not have a strict initial object
• does not have binary powers
• does not have countable powers
• does not have powers
• does not have finite powers
• does not have a coparametrized conatural numbers object
• is not infinitary coextensive
• is not countably coextensive
• does not have a strict terminal object
• does not have binary copowers
• does not have countable copowers
• does not have copowers
• does not have finite copowers
• is not cartesian closed
• is not countably distributive
• is not locally cartesian closed
• is not distributive
• is not extensive
• does not have products
• does not have countable products
• does not have finite products
• does not have binary products
• does not have ℵ₂-small powers
• does not have kernel pairs
• is not cocartesian coclosed
• is not countably codistributive
• is not locally cocartesian coclosed
• is not codistributive
• is not coextensive
• does not have coproducts
• does not have countable coproducts
• does not have finite coproducts
• does not have binary coproducts
• does not have ℵ₂-small copowers
• does not have cokernel pairs
• is not additive
• is not Grothendieck abelian
• does not have biproducts
• is not complete
• is not finitely complete
• does not have disjoint coproducts
• does not have disjoint finite coproducts
• is not infinitary distributive
• does not have cartesian filtered colimits
• does not satisfy CIP
• does not have ℵ₂-small products
• does not have pullbacks
• does not have coequalizers of kernel pairs
• is not an elementary topos
• is not a Grothendieck topos
• is not a pretopos
• is not a quasitopos
• is not co-Grothendieck abelian
• is not cocomplete
• is not finitely cocomplete
• does not have disjoint products
• does not have disjoint finite products
• is not infinitary codistributive
• does not have cocartesian cofiltered limits
• does not satisfy CSP
• does not have ℵ₂-small coproducts
• does not have pushouts
• does not have equalizers of cokernel pairs
• is not an elementary cotopos
• is not a Grothendieck cotopos
• is not a precotopos
• is not a quasicotopos
• is not Malcev
• is not unital
• is not locally finitely presentable
• is not locally ℵ₁-presentable
• is not locally presentable
• is not finitary algebraic
• is not abelian
• is not multi-complete
• does not have exact filtered colimits
• does not have wide pullbacks
• is not regular
• does not have pullback-stable regular epimorphisms
• does not have a subobject classifier
• does not have a regular subobject classifier
• is not total
• is not co-Malcev
• is not counital
• is not locally finitely copresentable
• is not locally ℵ₁-copresentable
• is not locally copresentable
• is not dually finitary algebraic
• is not multi-cocomplete
• does not have exact cofiltered limits
• does not have wide pushouts
• is not coregular
• does not have pushout-stable regular monomorphisms
• does not have a quotient object classifier
• does not have a regular quotient object classifier
• is not cototal
• is not locally multi-presentable
• is not locally finitely multi-presentable
• is not locally poly-presentable
• is not split abelian
• is not one-sorted finitary algebraic
• is not multi-algebraic
• does not have connected limits
• is not Barr-exact
• is not locally multi-copresentable
• is not locally finitely multi-copresentable
• is not locally poly-copresentable
• is not dually one-sorted finitary algebraic
• is not dually multi-algebraic
• does not have connected colimits
• is not Barr-coexact

*This also uses the deduced satisfied properties.