I saw two versions of semi-free DG modules $M$ over a small category $\mathcal{A}$, and I would like to know if they are equivalent or not.
In many Orlov's papers, a DG $\mathcal{A}$-module $M$ is called semi-free if it has a filtration $$o=\Phi_0\subset \Phi_1\subset \cdots =P$$ such that each quotient $\Phi_{i+1}/\Phi_i$ is free, that is, isomorphic to a direct sum of DG modules of the form $h^Y[n]$, where $Y\in \mathcal{A},\ n\in \mathbf{Z}$.
In Wai-Kit Yeung's paper, a DG $\mathcal{A}$-module $M$ is called semi-free if there exists an indexed set $\{x_\alpha\}_{\alpha\in S}$ of objects $x_{\alpha}\in \mathcal{A}$, together with an isomorphism $$M\cong \bigoplus_{\alpha\in S}h^{x_\alpha}$$ of underlying graded modules over the graded category after forgetting the differentials.
I like the second version more since it is more neat. However, I fail to see if these two are equivalent since we cannot guarantee that the quotient map $\Phi_{i+1}\to \Phi_{i+1}/\Phi_{i}$ splits and thus we cannot have the direct sum form of $M$ in the second definition.
Any reference or answers are welcome!