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