We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced by Guillén-Navarro-Pascual-Roig leading to a good calculation of the homotopy category in terms of (co)fibrant objects. Using Deligne's décalage, we show that the homotopy categories associated with the two notions of mixed Hodge complex introduced by Deligne and Beilinson respectively, are equivalent. The results provide a conceptual framework from which Beilinson's and Carlson's results on mixed Hodge complexes and extensions of mixed Hodge structures follow easily.
"Homotopy theory of mixed Hodge complexes." Tohoku Math. J. (2) 68 (3) 349 - 375, 2016. https://doi.org/10.2748/tmj/1474652264