Implication details
Claim: If a category is Cauchy complete and is retract-trivial and has zero morphisms, then it is thin.
Proof: For any object , the zero morphism is idempotent, hence a split idempotent by Cauchy completeness. Since it is retract-trivial, meaning all split idempotents are identities, we obtain . Now if are two morphisms, we conclude
Show 9 categories using this implication
- category of finite sets and bijections
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- category of finite sets and surjections
- category of finite groups