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 16 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
- category of finite abelian groups
- category of finite-dimensional vector spaces
- 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 finitely generated projective modules over the ring of dual numbers
- category of sets
- category of abelian sheaves
- category of simplicial sets
- walking splitting