Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$.

Under what assumptions is (the homotopy type of) $Y$ unique?

As has been pointed out below, the homotopy type of $Y$ being determined in general is far from true. But for connected $Y$, are the conditions where this is so?

Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$.

Under what assumptions is (the homotopy type of) $Y$ unique?

As has been pointed out below, the homotopy type of $Y$ being determined uniquely is far from true in general. But for connected $Y$, are there conditions we can impose that make it so?

Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$.

Under what assumptions is (the homotopy type of) $Y$ unique?

Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$.

Under what assumptions is (the homotopy type of) $Y$ unique?

As has been pointed out below, the homotopy type of $Y$ being determined in general is far from true. But for connected $Y$, are the conditions where this is so?