I am reading the QFT book by Weinberg. The first volume. On page 82, he discussed projective representation.
$$ U(T_2) U(T_1) = \exp(i \phi(T_2, T_1)) U(T_2 T_1 ) .\tag{2.7.1} $$
Here $\phi(T_2, T_1 )$ is the phase factor.
It is strange (at least to me) that he invoked the associativity condition
$$U(T_3) (U(T_2) U(T_1))= (U(T_3)U(T_2))U(T_1)$$
to impose the condition on $\phi$:
$$ \phi(T_2, T_1) + \phi(T_3, T_2 T_1) = \phi(T_3, T_2) + \phi(T_3 T_2, T_1). \tag{2.7.2}$$
Why is the associative law $$U(T_3) (U(T_2) U(T_1))= (U(T_3)U(T_2))U(T_1)~?$$ As you are now discussing projective representation, and you do not care about a phase factor, the associative law should be
$$ U(T_3) (U(T_2) U(T_1))\sim (U(T_3)U(T_2))U(T_1), $$
i.e., the two sides can differ up to a phase factor. There should be no requirement on $\phi$ at all.