There's certainly a homotopy-theoretic analogue.  A universal cover of a connected space X is (up to homotopy) a simply connected space X' and a map X' -> X which is an isomorphism on &pi;<sub>n</sub> for n >= 2.  We could next ask for a 2-connected cover X'' of X': a space X'' with &pi;<sub>k</sub>X'' = 0 for k <= 2 and a map X'' -> X' which is an isomorphism on &pi;<sub>n</sub> for n >= 3.  The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension 1--it will be a K(&pi;<sub>2</sub>X, 1).  (For the universal cover the fiber was the discrete space &pi;<sub>1</sub>X = K(&pi;<sub>1</sub>X, 0).)

An example is the Hopf fibration K(Z, 1) = S<sup>1</sup> -> S<sup>3</sup> -> S<sup>2</sup>.

Geometrically it's harder to see what's going on with the 2-connected cover than with the universal cover, because fibrations with fiber of the form K(G, 1) are harder to describe than fibrations with discrete fibers (covering spaces).