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A differential algebra with weight is an abstraction of both the derivation(weight zero) and the forward and backward difference operators (weight $\pm1$). In 2010 Loday established the Koszul duality for the operad ofdifferential algebras of weight zero. He did not treat the case of nonzeroweight, noting that new techniques are needed since the operad is no longerquadratic. This paper continues Loday's work and establishes the Koszul dualityin the case of nonzero weight. In the process, the minimal model and the Koszuldual homotopy cooperad of the operad governing differential algebras withweight are determined. As a consequence, a notion of homotopy differentialalgebras with weight is obtained and the deformation complex as well as its$L_\infty$-algebra structure for differential algebras with weight are deduced.

Koszul duality, minimal model and $L_\infty$-structure for differential algebras with weight