category of torsion-free abelian groups
- notation:
- objects: torsion-free abelian groups
- morphisms: group homomorphisms
- Related categories: , ,
- nLab Link
This is a typical example of a well-behaved additive category which is not abelian. It contains the category of free abelian groups.
Satisfied Properties
Assigned properties
- is locally small
- is complete
- is cocomplete
- is finitely accessible
- is preadditive
- has a cogenerator
- is regular
- is coregular
Deduced properties
- is locally finitely presentable
- is ℵ₁-accessible
- has filtered colimits
- is locally essentially small
- has zero morphisms
- has connected limits
- is finitely complete
- has equalizers
- has products
- is multi-complete
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is multi-cocomplete
- has a cogenerating set
- is inhabited
- is locally ℵ₁-presentable
- has exact filtered colimits
- is accessible
- has ℵ₁-filtered colimits
- is locally finitely multi-presentable
- has finite products
- has a multi-terminal object
- is strongly connected
- has coreflexive equalizers
- is Cauchy complete
- has kernels
- is filtered
- has directed colimits
- has sifted colimits
- has powers
- has ℵ₂-small products
- has wide pullbacks
- has finite coproducts
- has a multi-initial object
- has reflexive coequalizers
- has cokernels
- is cofiltered
- has cosifted limits
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- is locally presentable
- has a generating set
- is well-powered
- is locally multi-presentable
- is locally poly-presentable
- has filtered-colimit-stable monomorphisms
- is additive
- has quotients of congruences
- is semi-strongly connected
- has disjoint finite products
- has cartesian filtered colimits
- is sifted
- has countable products
- has ℵ₂-small powers
- has binary products
- has a terminal object
- has finite powers
- has cofiltered limits
- has pullbacks
- has coquotients of cocongruences
- has disjoint finite coproducts
- is cosifted
- has sequential colimits
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has an initial object
- has finite copowers
- has pushouts
- is Malcev
- is well-copowered
- has biproducts
- is connected
- is pointed
- has disjoint coproducts
- is ℵ₁-filtered
- has sequential limits
- has a generator
- has countable powers
- has binary powers
- is co-Malcev
- has disjoint products
- is ℵ₁-cofiltered
- has directed limits
- has ℵ₁-cofiltered limits
- has countable copowers
- has binary copowers
- is unital
- satisfies CIP
- is counital
- has cocartesian cofiltered limits
Unsatisfied Properties
Assigned properties
Deduced properties*
- is not discrete
- is not mono-regular
- is not gaunt
- is not direct
- does not have cofiltered-limit-stable epimorphisms
- is not epi-regular
- is not inverse
- is not self-dual
- is not trivial
- is not a groupoid
- is not normal
- does not have a subobject classifier
- is not subobject-trivial
- is not an elementary topos
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- is not conormal
- does not have a quotient object classifier
- does not have a natural numbers object
- is not abelian
- does not have effective congruences
- is not thin
- is not essentially discrete
- does not have a strict initial object
- is not finite
- does not have a regular subobject classifier
- is not a Grothendieck topos
- does not have effective cocongruences
- does not have a strict terminal object
- does not have a regular quotient object classifier
- is not countably distributive
- is not Grothendieck abelian
- is not split abelian
- is not multi-algebraic
- is not left cancellative
- is not cartesian closed
- is not core-thin
- is not Barr-exact
- is not distributive
- is not extensive
- is not locally finite
- is not one-way
- is not essentially small
- is not essentially countable
- is not cocartesian coclosed
- is not Barr-coexact
- is not codistributive
- is not coextensive
- is not locally copresentable
- is not locally strongly finitely presentable
- is not a generalized variety
- is not locally cartesian closed
- is not infinitary distributive
- is not infinitary extensive
- is not small
- is not countable
- is not a pretopos
- is not locally cocartesian coclosed
- is not countably codistributive
- is not infinitary coextensive
- is not finitary algebraic
- is not coaccessible
- is not infinitary codistributive
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: trivial group
- initial object: trivial group
- products: direct products
- coproducts: direct sums
Special morphisms
- isomorphisms: bijective group homomorphisms
- monomorphisms: injective group homomorphisms
- epimorphisms: homomorphisms such that is a torsion group
- regular monomorphisms: injective group homomorphisms such that is torsion-free, i.e., is the inclusion of a saturated subgroup
- regular epimorphisms: surjective group homomorphisms
Undistinguishable categories
These categories in the database currently have exactly the same properties as the category of torsion-free abelian groups. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.