The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p>0$. We introduce a category of cochain complexes equipped with an endomorphism $F$ of underlying graded abelian groups satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L η_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison for the $A Ω$-cohomology theory introduced in [BMS18].

158 pages. Final version. Contains a new section on the comparison with crystalline cohomology