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
Assigned properties
- is locally small
- is abelian
- is finitary algebraic
Deduced properties
- is additive
- has cokernels
- is conormal
- has kernels
- is normal
- is regular
- is locally strongly finitely presentable
- is well-copowered
- has a generator
- is locally essentially small
- is coregular
- is locally finitely presentable
- is cocomplete
- is a generalized variety
- has finite products
- is preadditive
- has biproducts
- is multi-algebraic
- is finitely complete
- has zero morphisms
- has a generating set
- is inhabited
- is mono-regular
- has finite coproducts
- is finitely cocomplete
- is epi-regular
- is Malcev
- is unital
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has sifted colimits
- is ℵ₁-accessible
- has filtered-colimit-stable monomorphisms
- is locally finitely multi-presentable
- is multi-cocomplete
- has effective congruences
- has equalizers
- has quotients of congruences
- is strongly connected
- is filtered
- is balanced
- has binary products
- has a terminal object
- has finite powers
- is co-Malcev
- is counital
- has connected colimits
- has coequalizers
- has coproducts
- is well-powered
- has coquotients of cocongruences
- has effective cocongruences
- is cofiltered
- has binary coproducts
- has an initial object
- has finite copowers
- is pointed
- is locally presentable
- is accessible
- has ℵ₁-filtered colimits
- has filtered colimits
- has connected limits
- is Grothendieck abelian
- is connected
- has a multi-terminal object
- is Barr-exact
- is semi-strongly connected
- has disjoint finite products
- has coreflexive equalizers
- is Cauchy complete
- has cartesian filtered colimits
- is sifted
- is ℵ₁-filtered
- has reflexive coequalizers
- has binary powers
- has pullbacks
- has a multi-initial object
- is Barr-coexact
- has disjoint finite coproducts
- is cosifted
- is ℵ₁-cofiltered
- has copowers
- has ℵ₂-small coproducts
- has binary copowers
- has pushouts
- has wide pushouts
- is complete
- is locally multi-presentable
- has a cogenerator
- has disjoint coproducts
- has directed colimits
- has wide pullbacks
- has cosifted limits
- has countable coproducts
- has ℵ₂-small copowers
- is locally poly-presentable
- has products
- is multi-complete
- has cofiltered limits
- has sequential colimits
- has a cogenerating set
- has countable copowers
- satisfies CIP
- has powers
- has ℵ₂-small products
- has disjoint products
- has cocartesian cofiltered limits
- has directed limits
- has ℵ₁-cofiltered limits
- has sequential limits
- has countable products
- has ℵ₂-small powers
- has countable powers
Unsatisfied Properties
Assigned properties
- is not skeletal
- is not split abelian
- does not satisfy CSP
Deduced properties*
- is not trivial
- is not discrete
- is not gaunt
- is not direct
- does not have cofiltered-limit-stable epimorphisms
- is not inverse
- is not self-dual
- does not have a natural numbers object
- is not thin
- is not core-connected
- is not essentially discrete
- is not a groupoid
- does not have a strict initial object
- does not have a regular subobject classifier
- is not regular-subobject-trivial
- is not regular-quotient-trivial
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- does not have a strict terminal object
- does not have a regular quotient object classifier
- is not countably distributive
- is not left cancellative
- is not cartesian closed
- is not distributive
- is not extensive
- is not finite
- does not have a subobject classifier
- is not subobject-trivial
- is not core-thin
- is not locally finite
- is not one-way
- is not essentially small
- is not essentially countable
- is not a quasitopos
- is not cocartesian coclosed
- is not codistributive
- is not coextensive
- does not have a quotient object classifier
- is not locally copresentable
- is not locally cartesian closed
- is not infinitary distributive
- is not infinitary extensive
- is not small
- is not countable
- is not an elementary topos
- is not a pretopos
- is not locally cocartesian coclosed
- is not countably codistributive
- is not infinitary coextensive
- is not a Grothendieck topos
- is not coaccessible
- is not infinitary codistributive
*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 morphisms
- monomorphisms: injective morphisms
- epimorphisms: surjective morphisms
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: surjective morphisms
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.
Functors
The database has stored 5 functors whose (co-)domain is the category of abelian groups.
Morphisms
The database has stored 1 morphism in the category of abelian groups.