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
- is connected
- has finite powers
- has binary powers
- is preadditive
- has zero morphisms
- is strongly connected
- has biproducts
- has finite coproducts
- has disjoint finite coproducts
- has coequalizers
- is epi-regular
- has equalizers
- has coreflexive equalizers
- is finitely complete
- has pullbacks
- is Cauchy complete
- is unital
- is mono-regular
- is balanced
- is regular
- is well-copowered
- is locally finitely presentable
- is locally presentable
- is locally ℵ₁-presentable
- is cocomplete
- is complete
- has connected limits
- has products
- has countable products
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- is well-powered
- has exact filtered colimits
- has filtered colimits
- has directed colimits
- is Malcev
- is pointed
- has an initial object
- has disjoint products
- has connected colimits
- has sifted colimits
- has reflexive coequalizers
- is finitely cocomplete
- has cosifted limits
- has binary coproducts
- has coproducts
- has disjoint coproducts
- is Grothendieck abelian
- has a cogenerator
- has countable coproducts
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- is counital
- has disjoint finite products
- has a cogenerating set
- is coregular
- is co-Malcev
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 infinitary distributive
- is not cartesian closed
- is not extensive
- is not infinitary extensive
- is not lextensive
- 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 finite
- is not essentially finite
- 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
- is not codistributive
- is not infinitary codistributive
- is not coextensive
- is not infinitary coextensive
- is not inverse
*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.