Implication Details
Claim: If a category is abelian, then it is regular.
Proof: In an abelian category, every epimorphism is regular, and epimorphisms are stable under pullbacks, see Mac Lane, Ch. VIII.
This implication has a dual.
Show 42 categories using this implication
- trivial category
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- category of small categories
- category of cochain complexes of abelian groups
- simplex category
- category of finite abelian groups
- category of finite sets of odd cardinality
- category of finite sets of cardinality a power of 3
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of Hausdorff spaces
- category of locally ringed spaces
- category of measurable spaces
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of pseudo-metric spaces with non-expansive maps
- category of partially ordered sets
- category of preordered sets
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of sequences of abelian groups
- indiscrete category of sets
- category of non-empty sets
- category of abelian sheaves
- category of topological spaces
- category of pointed topological spaces
- category of torsion abelian groups
- category of long transfinite sequences of abelian groups
- category of vector spaces
- category of countable-dimensional vector spaces
- category of large families of vector spaces which are mostly zero
- category of large vector spaces over a large field with a small basis
- forked commutative square
- category of graded abelian groups
- category of graded modules over a graded ring
- walking coreflexive pair
- walking isomorphism