Object as object in derived category

Question

while studying homological algebra, I got in touch with the notation, that for an object $X$ in a category $A$ and a chain complex $C$ in the derived category $D(A)$, the author considers $Hom(X,C)$ and its derived functor $RHom(X,C)$. However, it is not clear to me, how $X$ can be considered as an object in the derived category at all and could not find a canonical functor from $A$ to $D(A)$ (except considering it as a concentrated complex in degree zero, which most likely is not what I am searching for) in the literature so I am very thankful for any help with this probably very basic question.

Answer 1

Actually, one usually does implicitly treat an object in $A$ as a chain complex concentrated in degree zero. In some sense, the derived category $D(A)$ can be viewed as enlarging $A$ to allow derived operations; e.g. taking the mapping cone of a morphism $X \to Y$, if you view them as complexes concentrated in degree zero, results in the two-term complex $\ldots \to 0 \to X \to \underline{Y} \to 0 \to \ldots $ (with the degree 0 term underlined).

Note, incidentally, that $\hom(X,C)$ considers $C$ as a chain complex, not as an object of the derived category; you might get a different hom-complex if you replace $C$ with a quasi-isomorphic complex. In some sense, this problem is the whole point of the $\mathbb{R}{\hom}(X,C)$ construction, and why you might restrict yourself to considering $C$ being a complex of injective objects.