abelian
A category is abelian if it is additive, every morphism has a kernel and a cokernel, and every monomorphism and epimorphism is normal. Equivalently, it is additive, has equalizers and coequalizers, and it is mono-regular and epi-regular. As opposed to other types of categories (such as monoidal categories), being abelian turns out to be a mere property.
Relevant implications
- abelian is equivalent to additive andcoequalizers andepi-regular andequalizers andmono-regular
- abelian implies coregular
- abelian implies regular
- Grothendieck abelian is equivalent to abelian andcoproducts andexact filtered colimits andgenerator
- split abelian implies abelian
Examples
There are 9 categories with this property.
- category of abelian groups
- category of abelian sheaves
- category of finite abelian groups
- category of finitely generated abelian groups
- category of left modules over a division ring
- category of left modules over a ring
- category of vector spaces
- trivial category
- walking isomorphism
Counterexamples
There are 56 categories without this property.
- category of algebras
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of fields
- category of finite orders
- category of finite sets
- category of finite sets and bijections
- category of finite sets and injections
- category of finite sets and surjections
- category of free abelian groups
- category of groups
- category of Hausdorff spaces
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of monoids
- category of non-empty sets
- category of pairs of sets
- category of pointed sets
- category of posets
- category of prosets
- category of rings
- category of rngs
- category of schemes
- category of sets
- category of sets and relations
- category of sheaves
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category on two objects
- dual of the category of sets
- empty category
- poset [0,1]
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- walking commutative square
- walking composable pair
- walking fork
- walking morphism
- walking parallel pair of morphisms
- walking span
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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