I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and Amnon Neeman and can't understand at all even some of the notation. From what I've gathered, the resolution of an unbounded complex $X$ is $M$ in:

$X =$ $\underleftarrow{lim_{n}} (X_{-n}) \xrightarrow{\alpha} \underleftarrow{holim_{n}} (X_{-n}) \xrightarrow{\beta} M = \underleftarrow{holim_{n}} (I_{-n})$,

where $\alpha$ and $\beta$ are quasi-isomorphisms.

But what's the homotopy limit in this context? The thing I can relate it to is this:

A homotopy limit of a sequence of maps

$X_{0} \xrightarrow{f_0} X_{1} \xrightarrow{f_1} X_{2} \rightarrow \cdots$

is defined as the subspace of the product of $X_0$ with $X_{i}\times\hom(\Delta[1],X_{i})$, $i \geq 1$, of elements $(x_0,${$(x_i,\gamma_i)$}$_{0 \leq i \leq n}$) such that

$\gamma_i(0) = f(x_{i-1})$,

$\gamma_i(1) = x_i$.

This is completely over my head, as someone coming from Cartan-Eilenberg resolutions being the resolutions of bounded complexes I was expecting to see a bicomplex, I have no idea how to interpret that, does anyone know? Be nice please, I'm still learning.