Universal Homotopy Theories and Associated Homological Algebras

Let $\mathcal{C}$ be a small category. For every commutative ring $R$ with unity, we associate an $R-\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$ abelian category with the universal homotopy category of $\mathcal{C}$, where we can do the corresponding homological algebra.