For a pair of snc models ${\cal X}^{\prime}\geq{\cal X}$ as above we say that ${\cal X}^{\prime}$ is obtained from ${\cal X}$ by a simple blow-up. If $Y=D_{J}$ we say that we have a simple blow-up of the first type. Otherwise (when $\dim(Y)<\dim(D_{J})$), we have a simple blow-up of the second type.