Implication Details

Assumptions: disjoint finite coproducts, strict terminal object

Conclusions: thin

Reason: Let $1$ be the strict terminal object, and let $A$ be any object. Then $1 \to A + 1$ is an isomorphism, since $1$ is strict. Also, $A \to A + 1$ is a monomorphism by assumption. It follows that the unique morphism $u : A \to 1$ is a monomorphism. For all $f,g : B \to A$ we have $uf = ug$ (since $1$ is terminal), hence $f = g$ .

Show 28 categories using this implication

category of finitely generated abelian groups
category of algebras
category of Banach spaces with linear contractions
category of commutative algebras
category of commutative monoids
category of commutative rings
category of compact Hausdorff spaces
category of finite abelian groups
category of free abelian groups
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 sets and relations
category of rings
category of rngs
category of schemes
category of semigroups
category of countable sets
category of sets with finite-to-one maps
category of pointed sets
category of topological spaces
category of pointed topological spaces
category of Z-functors