A map lifts to a cover exactly when its fundamental group lands in the covering subgroup.Suggest a slogan
Let \(p\colon(\tilde X, \tilde x_0)\to(X, x_0)\) be a covering space and \(f\colon(Y, y_0)\to(X, x_0)\) a map with \(Y\) path-connected and locally path-connected. Then a lift \(\tilde f\colon(Y, y_0)\to(\tilde X, \tilde x_0)\) with \(p\circ\tilde f = f\) exists if and only if \(f_*(\pi_1(Y, y_0)) \subseteq p_*(\pi_1 (\tilde X, \tilde x_0))\) [@Hat02]:
If \(Y\) is connected, two lifts of \(f\) that agree at one point are equal [@Hat02].