Implication Details
Claim: If a category has disjoint finite coproducts and has a strict terminal object, then it is thin.
Proof: Let be the strict terminal object, and let be any object. Then is an isomorphism, since is strict. Also, is a monomorphism by assumption. It follows that the unique morphism is a monomorphism. For all we have (since is terminal), hence .
This implication has a dual.
Show 43 categories using this implication
- category of finitely generated abelian groups
- category of algebras
- category of Banach spaces with linear contractions
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of compact Hausdorff spaces
- 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 free abelian groups
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of groups
- category of countable groups
- category of Hausdorff spaces
- category of locally ringed spaces
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with ∞ allowed
- category of monoids
- category of finitely generated projective modules over the ring of dual numbers
- category of quivers with finite components
- category of sets and relations
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of pointed sets
- category of countable sets
- category of sets with finite-to-one maps
- category of topological spaces
- category of pointed topological spaces
- category of long transfinite sequences of abelian groups
- category of uniform spaces
- category of countable-dimensional vector spaces
- category of large families of vector spaces which are mostly zero
- category of Z-functors
- cocompletion of a discrete–pair join