Why are projective representations of a group classified by the second cohomology group?

Question

I'm reading about the classification of bosonic SPT's, and I came across this statement: projective representations, where $v(g_1)v(g_2)=\alpha(g_1,g_2)v(g_1g_2)$, $v(g_1)$ being the transformation corresponding to $g_1$ and $\alpha$ being a $U(1)$ phase, can be classified according to the elements of the second cohomology group $H^2(G,U(1))$. Is there an intuitive way of understanding this?

Answer 1

The argument is standard and can be found, for example, here. Beyond just checking that everything works out, it depends on what intuitions you have about cohomology. You might be interested in reading the blog posts immediately before and after the one above, which are much more interesting.