Implication details
Claim: If a category is core-connected and is locally finite, then it is retract-trivial.
Proof: A core-connected locally finite category is equivalent to for a finite monoid . A finite monoid satisfies the property . In fact, when holds, the map is surjective, hence injective. Thus, implies .
Show 12 categories using this implication
- trivial category
- delooping of the bicyclic monoid
- delooping of the double bicyclic monoid
- simplex category
- category of finite sets of odd cardinality
- category of finite sets of cardinality a power of 3
- category of finite-dimensional vector spaces [finite field]
- category of countably infinite sets
- indiscrete category of sets
- walking coreflexive pair
- walking isomorphism
- walking splitting