I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
Suppose you have topological spaces $X,Y, Z$, and morphisms $\phi: X \to Y$ and $p : Z \to Y$. Then one raises the question if the map $\phi$ is liftable, i.e. the existence of a map $f : X \to Z$ such that $\phi = p \circ f$. (Probably I have forgotten some assumptions on the spaces but that's not the point here for me)
One can prove that this lifting problem has a solution if and only if the "corresponding" lifting problem regarding the fundamental groups has a solution. (I mean, consider the corresponding fundamental groups and induced group homomorphism and ask the same question)
Our teacher told us (informally) that typically the "topological lifting problem" is difficult where the "algebraic lifting problem" can be used to prove the former one.
My question is the following: Has there been some study (I personally guess so) to solve the "algebraic lifting problem" using a translation to a "lifting problem" living in another category.
Hence, let $G,H,T$ be groups, morphisms $f : G \to H$ and $p : T \to H$. We want to lift $f$ to $T$. We may also assume that $p$ is surjective (i.e. $H$ is a quotient of $T$).
I thought of translating this to group algebras or $C^*$-group algebras.
