Implication Details

Assumptions: sequential colimits

Conclusions: Cauchy complete

Reason: Assume that e:X→Xe : X \to X is an idempotent morphism. Consider the sequence X→eX→eX→⋯X \xrightarrow{e} X \xrightarrow{e} X \to \cdots. A cocone under this sequence is a family of morphisms fn:X→Yf_n : X \to Y satisfying fn=fn+1ef_n = f_{n+1} e. Then fn=fn+1e=fn+2e2=fn+2e=fn+1f_n = f_{n+1} e = f_{n+2} e^2 = f_{n+2} e = f_{n+1} shows that all the morphisms are equal. Thus, a cocone is the same as a morphism f0:X→Yf_0 : X \to Y with f0=f0ef_0 = f_0 e, meaning it coequalizes idX,e:X⇉X\mathrm{id}_X,e : X \rightrightarrows X. Hence, if a colimit exists, ee splits.