nLab
twisted module of homomorphisms

Let $C$ be a dg-coalgebra, $A$ a dg-algebra, $N$ a left $C$-dg-comodule with coaction ${\delta }_N:N\to C\otimes N$, $P$ a left $A$-dg-module with action $m_P:A\otimes P\to P$ and $\tau :C\to A$ a twisting cochain. The twisted module of homomorphisms ${\mathrm{Hom}}_{\tau }(N,P)$ is a chain complex which as a graded module coincides with the ordinary module of homomorphisms of the underlying chain complex $\mathrm{Hom}(N,P)$, and with the differential $d_{\tau }$ given by

$$d_{\tau }(f)=d(f)+m_P\circ (\tau \otimes f)\circ {\delta }_N,$$d_\tau(f) = d(f) + m_P\circ(\tau\otimes f)\circ\delta_N,

where $f\in \mathrm{Hom}(N,P)$.