Definition of a lift in algebraic topology

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.

• Are you aware of the concept of convering spaces ? May 5 '14 at 19:10
• Yes. ${}{}{}{}$ May 5 '14 at 20:05

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 $f$.
• Need $\widetilde{X}$ and $\rho$ form a covering space for this definition to be satisfied? May 6 '14 at 0:41
• Not at all, we can ask this question for any $\tilde{X}$ and $\rho$ whatsoever. However, in many naturally occurring situations, $\rho$ does have properties resembling a covering space (specifically, it is often a fibration). May 6 '14 at 1:04
• So if $f: X \rightarrow Y$, all we need to lift $f$ is another set $\widetilde{X}$, mappings $\widetilde{f}: X \rightarrow \widetilde{X}$ and $\rho: \widetilde{X} \rightarrow Y$, and the commutative property that $\rho \circ \widetilde{f} = f$ for us to have a lift. May 6 '14 at 15:10
• Yes, then by definition, $\tilde{f}$ is a lift of $f$ through $\rho$. Of course, we rarely want just any lift, but one through a specified map $\rho$. May 6 '14 at 17:05