Let $(C,\mathcal{W},\mathcal{C},\mathcal{F})$ be a model category. The homotopy category is defined as the localization of category $C$ with respect to $\mathcal{W}$, denoted by $C[\mathcal{W}^{-1}]$ or $HoC$. Let $\mathcal{C}_c$ denote the full subcatogory of cofibrant objects of $C$.
And then I saw the expression $HoC_c$, meaning the homotopy category of $C_c$? But $\mathcal{W}$ is not contained in $Mor(C_c)$, right? I am really annoyed by this $HoC_c$ thing. Can anyone explain?