Does anyone know a textbook or even some notes online where I can find the definition of DG-injective chain complex?
I tried to type it on google but I only find articles or papers about DG-injective complexes, so they don't start from the definitions.
It seems the terminology is not quite standard. As far as I can tell, the definition is due to Avramov, Foxby, and Halperin [unpublished, Differential graded homological algebra]. It is stated in [Avramov and Foxby, 1991, Homological dimensions of unbounded complexes] as follows:
A chain complex $C$ is dg-injective if, for every injective chain complex homomorphism $f : A \to B$ that is a quasi-isomorphism, $\textrm{Hom} (f, C) : \textrm{Hom} (B, C) \to \textrm{Hom} (A, C)$ is a quasi-isomorphism. Here, $\textrm{Hom}$ denotes the total hom complex, i.e. $$\textrm{Hom} (A, C)_n = \prod_{i \in \mathbb{Z}} \textrm{Hom} (A_i, C_{i + n})$$ As it turns out, the following are equivalent for a chain complex $C$:
$C$ is dg-injective.
Every $C_n$ is injective and $\textrm{Hom} (-, C)$ preserves quasi-isomorphisms.
Every $C_n$ is injective and $\textrm{Hom} (-, C)$ preserves acyclic complexes.
Thus, dg-injective chain complexes are examples of what nLab calls homotopically injective objects.
