Implication Details

Claim: If a category has biproducts and has filtered colimits and has filtered-colimit-stable monomorphisms and has products, then it satisfies CIP.

Proof: Let (Xi)i∈I(X_i)_{i \in I} be a family of objects. For every finite subset E⊆IE \subseteq I the canonical morphism ∐i∈EXi=∏i∈EXi→∏i∈IXi\coprod_{i \in E} X_i = \prod_{i \in E} X_i \to \prod_{i \in I} X_i is a (split) monomorphism. Hence, their colimit is also a monomorphism, which is the canonical morphism ∐i∈IXi→∏i∈IXi\coprod_{i \in I} X_i \to \prod_{i \in I} X_i.

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