# Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ following Bousfield and Kan; more precisely, $\hocolim X = \operatorname{diag} T_{\bullet}$ where $T_{\bullet}$ is the bisimplicial set defined below: $$T_n = \coprod_{(j_0, \ldots, j_n)} \mathcal{J} (j_{n-1}, j_n) \times \cdots \times \mathcal{J} (j_0, j_1) \times X (j_0)$$

Question. I am led to believe that there are spectral sequences relating the (co)homology of the individual simplicial sets $X (j)$ and the (co)homology of $\hocolim X$. What is the precise statement, and where is it written in the literature?

For convenience, I quote Dugger:

Cohomology spectral sequence. Let $E$ be a spectrum. There is a spectral sequence with $$E_2^{p, q} = H^p (\mathcal{J}^\mathrm{op}, E^q (X))$$ and differentials $d : E_r^{p, q} \to E_r^{p + r, q - r + 1}$, converging to $E^{p - q} (\hocolim X)$.

Homology spectral sequence. Let $E$ be a spectrum. There is a spectral sequence with $$E^2_{p, q} = H_q (\mathcal{J}, E_p (X))$$ and differentials $d : E^r_{p, q} \to E^r_{p + r - 1, q - r}$, converging to $E_{p + q} (\hocolim X)$.

Specifically, I want to know whether the filtrations on $E^n (\hocolim X)$ and $E_n (\hocolim X)$ are exhaustive, canonically bounded, etc. (Actually, I'm not completely sure what Dugger means by $H_q (\mathcal{J}, -)$; I'm guessing he means the left derived functors of $\varinjlim$, by analogy with $H^q (\mathcal{J}^\mathrm{op}, -)$ being the right derived functors of $\varprojlim$.)

If it simplifies things, I'm only interested in the case where $E$ is an Eilenberg–MacLane spectrum.