delooping of an infinite countable group

Satisfied properties

Assigned properties

• is countable

Properties from parent

• is small
• is a groupoid
• is core-connected
• is skeletal

Deduced properties

• is strongly connected
• has an extremal generator
• has directed limits
• is left cancellative
• is mono-regular
• has pullbacks
• is self-dual
• is well-powered
• is locally cartesian closed
• is essentially small
• is locally small
• is essentially countable
• is subobject-trivial
• has an extremal cogenerator
• has directed colimits
• is epi-regular
• has pushouts
• is right cancellative
• is well-copowered
• is locally cocartesian coclosed
• is quotient-trivial
• is a generalized variety
• is finitely accessible
• is Cauchy complete
• has coreflexive equalizers
• has effective cocongruences
• has effective congruences
• has reflexive coequalizers
• is semi-strongly connected
• has filtered colimits
• has sequential limits
• has an extremal generating collection
• has a generator
• is balanced
• has kernel pairs
• is locally essentially small
• is regular-subobject-trivial
• is finitely coaccessible
• has cofiltered limits
• has sequential colimits
• has a cogenerator
• has an extremal cogenerating collection
• has cokernel pairs
• is regular-quotient-trivial
• has filtered-colimit-stable monomorphisms
• is ℵ₁-accessible
• is accessible
• has sifted colimits
• has quotients of congruences
• is connected
• has ℵ₁-filtered colimits
• has a generating collection
• is inhabited
• has wide pullbacks
• has pullback-stable regular epimorphisms
• has cofiltered-limit-stable epimorphisms
• is ℵ₁-coaccessible
• is coaccessible
• has coquotients of cocongruences
• has ℵ₁-cofiltered limits
• has cosifted limits
• has a cogenerating collection
• has wide pushouts
• has pushout-stable regular monomorphisms
• is locally poly-presentable
• has coequalizers of kernel pairs
• is concretizable
• has equalizers of cokernel pairs

Unsatisfied properties

Assigned properties

• is not locally finite

Properties from parent

• is not thin

Deduced properties*

• does not have zero morphisms
• does not have binary copowers
• is not essentially discrete
• is not sifted
• does not have a multi-terminal object
• is not essentially finite
• does not have equalizers
• is not core-thin
• does not have powers
• does not have countable powers
• does not have binary powers
• is not cosifted
• does not have a multi-initial object
• does not have coequalizers
• does not have copowers
• does not have countable copowers
• is not preadditive
• does not have biproducts
• is not complete
• is not finitely complete
• is not multi-complete
• does not have a terminal object
• is not trivial
• is not discrete
• does not have binary products
• does not have kernels
• does not satisfy CIP
• is not filtered
• does not have binary coproducts
• is not pointed
• is not normal
• does not have products
• does not have countable products
• does not have ℵ₂-small powers
• does not have finite powers
• does not have finite copowers
• does not have connected limits
• is not finite
• is not gaunt
• is not one-way
• is not cocomplete
• is not finitely cocomplete
• is not multi-cocomplete
• does not have an initial object
• does not have cokernels
• does not satisfy CSP
• is not cofiltered
• is not conormal
• does not have coproducts
• does not have countable coproducts
• does not have ℵ₂-small copowers
• does not have connected colimits
• is not Malcev
• is not unital
• does not have a natural numbers object
• 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 additive
• is not abelian
• is not Grothendieck abelian
• is not multi-algebraic
• does not have disjoint coproducts
• is not infinitary distributive
• is not countably distributive
• does not have exact filtered colimits
• is not countably extensive
• is not infinitary extensive
• is not ℵ₁-filtered
• does not have a strict initial object
• does not have ℵ₂-small products
• does not have finite products
• is not regular
• does not have a subobject classifier
• does not have a regular subobject classifier
• does not have disjoint finite coproducts
• is not direct
• is not an elementary topos
• is not a Grothendieck topos
• is not a quasitopos
• is not total
• is not co-Malcev
• is not counital
• is not locally finitely copresentable
• is not locally ℵ₁-copresentable
• is not locally copresentable
• does not have disjoint products
• is not infinitary codistributive
• is not countably codistributive
• does not have exact cofiltered limits
• is not countably coextensive
• is not infinitary coextensive
• is not ℵ₁-cofiltered
• does not have a strict terminal object
• does not have ℵ₂-small coproducts
• does not have finite coproducts
• is not coregular
• does not have a quotient object classifier
• does not have a regular quotient object classifier
• does not have disjoint finite products
• is not inverse
• is not cototal
• does not have a parametrized natural numbers object
• is not split abelian
• is not one-sorted finitary algebraic
• is not cartesian closed
• is not distributive
• does not have cartesian filtered colimits
• is not extensive
• is not Barr-exact
• is not cocartesian coclosed
• is not codistributive
• does not have cocartesian cofiltered limits
• is not coextensive
• is not Barr-coexact
• is not a pretopos

*This also uses the deduced satisfied properties.