This question is similar to here, but I was hoping for a concrete theorem statement surrounding the obstruction cocycle for non-single spaces.

I hope for a theorem like this:

Let $ A subset X $ such as $ pi_1 (A) = pi_1 (X) $ and $ f: A to Y $ to be a function. leasing $ pi_n (Y) $ be a $ mathbb Z ( pi_1 (A)) $ module, where $ pi_1 (A) $ acts via $ f _ * $ and the usual action of $ pi_1 (Y) $. Assume that $ H ^ * (X, A, pi_n (Y)) = 0 $ for everyone $ n in mathbb N $. There is then an extension of $ f $ to all $ X $. More generally if $ pi_1 (X) = pi_1 (A) / N $ and $ N subset ker f _ * $, then if $ H ^ * (X, A, pi_n (A) / N) = 0 $, there is an extension.

The slightly awkward generalization is due to my desire to prove universal property for construction Quillen & # 39; s more presented in these notes, prop 1.1.2.

A naive assumption on my part would be to apply the usual obstruction theorem to universal coverage of $ Y $, then if we use local coefficients, we can make sure that $ f: A à tilde {Y} $ is a lift of $ f: A to Y $, but I'm not completely sure.