Let $\mathcal{A}$ be an abelian category, and let $Ch_*(\mathcal{A})$ denote the associated category of chain complexes. We define projective resolutions in this general case:
For an object $X \in Ob(\mathcal{A})$, a projective resolution of $X$ is an object $C_* \in Ob(Ch_*(\mathcal{A}))$, with $C_j = 0$ for $j < 0$, equipped with a quasi-isomorphism $$C_* \xrightarrow{\cong} X[0],$$
where $X[0]$ is the chain complex with $X$ in degree zero and $0$ in all other degrees, such that $C_n$ is projective for all $n \in > \mathbb{N}$.
We define the length of a projective resolution $C_*$ to be $n \in > \mathbb{N}$ such that $C_n \neq 0$, and $C_i = 0$ for $i > n$. If no such $n$ exists, we say that the length is $\infty$.
I am interested in the definition of projective dimension in such a general case.
For example, let $R$ be a commutative ring. Taking $\mathcal{A} = \mathbf{RMod}$, the category of $R$-modules, we define the projective dimension of $M \in Ob(\mathbf{RMod}$) to be the minimal length over all finite length projective resolutions of $M$. If there exist no finite resolutions, we say that the projective dimension of $M$ is $\infty$.
I can find no reference to a notion of projective dimension in the general case. In particular, I would like to know whether there exists a notion of projective dimension in the case that $\mathcal{A} = Ch_*(\mathbf{RMod})$; that is, when the objects of $\mathcal{A}$ are themselves chain complexes. A projective resolution of such would then be a double complex.
It seems like it is easy to generalise the definition of projective dimension to such a case, and in fact to any abelian category; the projective dimension is the minimal length over all projective resolutions (and is $\infty$ if there are no finite projective resolutions). We would require that $\mathcal{A}$ has enough projectives, so that every object has a projective resolution, but that's all.
Are there any issues with this definition of projective dimension for general $\mathcal{A}$? It seems like it might be a useful invariant, potentially for example in investigating (cellular) chain complexes of topological spaces.