Necessary and sufficient conditions for complements to presilting objects in triangulated categories are established via co-t-structures, plus an equivalence characterizing silting-discrete algebras.
Keller, Deriving DG categories , Ann.\ Sci.\ \'Ecole Norm.\ Sup.\ 27 (1994), no.\ 1, 63--102
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Fishing for complements
Necessary and sufficient conditions for complements to presilting objects in triangulated categories are established via co-t-structures, plus an equivalence characterizing silting-discrete algebras.