Implication details
Claim: If a category has kernel pairs and is retract-trivial, then it is left cancellative.
Proof: Let be a morphism with kernel pair . The diagonal is a split monomorphism. By assumption on the category, it is an isomorphism. Therefore, is a pullback, so is a monomorphism.
This implication has a dual.
Show 15 categories using this implication
- category of Banach spaces with linear contractions
- category of commutative algebras
- category of commutative rings
- category of compact Hausdorff spaces
- category of finite sets and surjections
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of groups
- category of countable groups
- category of Hausdorff spaces
- partially ordered set of extended natural numbers
- category of connected sequences of sets
- category of torsion abelian groups
- walking fork
- walking idempotent