Implication Details
Claim: If a category is a groupoid and has a multi-terminal object, then it is thin.
Proof: Let be a parallel pair of morphisms. Since the category has a multi-terminal object, the connected component containing and has a terminal object. But since the category is a groupoid, both and are terminal objects in the connected component, hence .
Show 12 categories using this implication
- category of finite abelian groups
- category of finite sets
- category of finite sets and bijections
- 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 M-sets
- category of sets
- category of simplicial sets
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- walking splitting