Implication Details
Claim: If a category has equalizers, then it is Cauchy complete.
Proof: If is an idempotent, then the equalizer of provides a splitting of .
This implication has a dual.
Show 55 categories using this implication
- category of large families of abelian groups
- category of algebras
- category of sets equipped with a binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of commutative algebras
- category of commutative rings
- category of cochain complexes of abelian groups
- category of compact Hausdorff spaces
- category of coproducts of Euclidean spaces
- category of F(I)-sets
- category of filtered vector spaces
- category of finite ordered sets
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite sets of cardinality a power of 3
- category of free abelian groups
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of groups
- category of countable groups
- 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 continuous maps
- category of monoids
- category of sets with a distinguished subset
- category of pseudo-metric spaces with non-expansive maps
- category of partially ordered sets without isolated points
- category of quivers with finite components
- category of sets and relations
- category of rings
- category of rngs
- category of sequences of abelian groups
- category of pointed sets
- category of countable sets
- category of large families of sets
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of sets with finite-to-one maps
- category of empty-or-finite pairs of sets
- category of topological spaces
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of long transfinite sequences of abelian groups
- category of uniform 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
- category of Z-functors
- category of graded abelian groups
- category of graded modules over a graded ring
- walking idempotent
- walking splitting