Suppose that $P\subset Q\subset\mathbb{R}^{d}$ are polytopes such that $P$ is contained in the relative interior of some face of $Q$. Then the result of pulling $P$ is by definition the coherent subdivision $\operatorname{pull}_{P}(Q)$ of $Q$ which is induced by the heights $1$ on $P$, and $0$ on all faces of $Q$ that do not contain $P$.