Abstract: We will consider a finite subgroup $\Gamma$ of $SL_2(\mathbb{C})$. The quotient $Y = \mathbb{C}^2 / \Gamma$ is one of the simplest examples of a singular algebraic variety. I will discuss an equivalence between the derived category of coherent sheaves of a minimal resolution of $Y$ and the $\Gamma$-equivariant derived category of $\mathbb{C}^2$. This is connected to and has inspired many developments in geometry and representation theory. No prior knowledge of derived categories is required.