category of countable groups
A group is called countable if its underlying set is countable. In particular, every finite group is countable, but also every finitely generated group is countable.
Satisfied Properties
Assigned properties
- is locally small
- is essentially small
- is pointed
- has a generator
- has finite products
- has equalizers
- has coequalizers
- has countable coproducts
- is mono-regular
- is conormal
- is Malcev
- is regular
- has effective congruences
- has effective cocongruences
Deduced properties
- is finitely complete
- is unital
- has quotients of congruences
- is Barr-exact
- has coreflexive equalizers
- is Cauchy complete
- has a generating set
- is inhabited
- has an initial object
- has zero morphisms
- is balanced
- has binary products
- has a terminal object
- has finite powers
- is locally essentially small
- is well-copowered
- is well-powered
- has reflexive coequalizers
- has sequential colimits
- is epi-regular
- has countable copowers
- has finite coproducts
- has a cogenerating set
- is accessible
- is connected
- has a multi-terminal object
- is strongly connected
- has kernels
- is filtered
- has binary powers
- has pullbacks
- is coaccessible
- is finitely cocomplete
- has a multi-initial object
- has coquotients of cocongruences
- has cokernels
- is cosifted
- is cofiltered
- has binary coproducts
- has finite copowers
- is semi-strongly connected
- has disjoint finite products
- is sifted
- has disjoint finite coproducts
- has binary copowers
- has pushouts
Unsatisfied Properties
Assigned properties
- is not skeletal
- is not small
- is not normal
- is not counital
- does not have countable powers
- does not have a regular quotient object classifier
- is not coregular
- does not have a cogenerator
- is not ℵ₁-accessible
Deduced properties*
- is not locally ℵ₁-presentable
- is not finitely accessible
- is not abelian
- is not Grothendieck abelian
- is not a generalized variety
- is not cartesian closed
- is not additive
- is not discrete
- is not preadditive
- does not have countable products
- does not have powers
- does not have sequential limits
- does not have a subobject classifier
- is not gaunt
- is not direct
- is not an elementary topos
- is not a Grothendieck topos
- is not co-Malcev
- does not have biproducts
- is not locally cocartesian coclosed
- is not Barr-coexact
- does not have products
- is not left cancellative
- does not have a quotient object classifier
- is not thin
- is not inverse
- is not self-dual
- is not locally finitely presentable
- is not essentially finite
- is not locally strongly finitely presentable
- is not locally finitely multi-presentable
- is not split abelian
- is not multi-algebraic
- is not right cancellative
- is not locally cartesian closed
- is not complete
- is not extensive
- is not trivial
- is not essentially discrete
- does not have a strict terminal object
- does not satisfy CIP
- does not have directed limits
- is not a groupoid
- does not have cofiltered limits
- does not have a regular subobject classifier
- is not subobject-trivial
- is not core-thin
- is not one-way
- does not have disjoint products
- does not have a strict initial object
- is not infinitary codistributive
- is not countably codistributive
- does not satisfy CSP
- is not infinitary coextensive
- is not quotient-trivial
- is not locally finite
- does not have copowers
- is not essentially countable
- does not have a natural numbers object
- is not locally presentable
- is not finitary algebraic
- does not have wide pullbacks
- is not multi-complete
- is not distributive
- is not infinitary extensive
- is not finite
- is not countable
- is not locally copresentable
- is not cocartesian coclosed
- is not codistributive
- does not have exact cofiltered limits
- does not have cocartesian cofiltered limits
- does not have cofiltered-limit-stable epimorphisms
- is not coextensive
- does not have cosifted limits
- does not have coproducts
- does not have filtered colimits
- is not countably distributive
- is not cocomplete
- is not locally poly-presentable
- does not have disjoint coproducts
- 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 have directed colimits
- does not have sifted colimits
- does not have connected limits
- does not have wide pushouts
- is not locally multi-presentable
- does not have connected colimits
- is not multi-cocomplete
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: trivial group
- initial object: trivial group
- products: [finite case] direct products with pointwise operations
- coproducts: [countable case] free products
Special morphisms
- isomorphisms: bijective homomorphisms
- monomorphisms: injective homomorphisms
- epimorphisms: surjective homomorphisms
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms