Let \(p\colon E\to B\) be a continuous map and \(Y\) a topological space. The map \(p\) has the homotopy lifting property with respect to \(Y\) if for every homotopy \(g_t\colon Y\to B\), \(t\in[0,1]\), and every continuous \(\tilde g_0\colon Y\to E\) with \(p\circ\tilde g_0 = g_0\), there is a homotopy \(\tilde g_t\colon Y\to E\) starting at \(\tilde g_0\) with \(p\circ\tilde g_t = g_t\) for all \(t\). The map \(p\) is a fibration if it has the homotopy lifting property with respect to every space \(Y\).
A covering space \(p\colon\tilde X\to X\) has the homotopy lifting property with respect to every space \(Y\), and the lifted homotopy \(\tilde g_t\) starting at a given \(\tilde g_0\) is unique.
See [@Hat02].