Definition of a lift in algebraic topology

Question

Definition: A lift of a map $f: X \rightarrow Y$ is a map $\widetilde{f}: X\rightarrow \widetilde{X}$ s.t. $\rho \widetilde{f} = f$.

Question: What here is meant by the map $\rho$? In my text (Hatcher) it doesn't seem to be explicitly defined.

Answer 1

We talk about lifting a map $f$ through another map $\rho$. The map $\rho$ is therefore a part of the data, together with the spaces $X,\tilde{X}$ and the map $f$.