Implication Details

Claim: If a category has a cogenerating collection and has disjoint coproducts, then it has a cogenerator.

Proof: Assume that $S$ is a cogenerating collection and let $Q \coloneqq \coprod_{X \in S} X$ . For $X \in S$ we have a monomorphism $i_X : X \to Q$ . If $f,g : A \rightrightarrows B$ are two distinct morphisms, there is some $X \in S$ and a morphism $h : B \to X$ with $hf \neq hg$ . Hence, $i_X h f \neq i_X h g$ . This proves that $Q$ is a cogenerator.

This implication has a dual.

Show 12 categories using this implication

empty category
category of finitely generated abelian groups
category of finite sets and bijections
category of commutative monoids
category of F(I)-sets
category of finite sets and injections
category of finite groups
category of countable groups
category of quivers with finite components
category of long transfinite sequences of abelian groups
category of large families of vector spaces which are mostly zero
category of finite Z-sets