Let $f: C_\bullet \to D_\bullet$ be chain map between chain complexes. What does it mean for $f$ to be a split injection? And a split surjection?
I am aware of the definitions of split monomorphism and split epimorphism in an arbitrary category. However, is there a more intuitive definition of split injection or of split surjection in the category of chain complexes of modules (perhaps some definition that involves a direct sum)? I cannot seem to find a good reference.