Fix some standard Polish space, e.g. Baire's space. It's a simple observation that every $\Delta^1_1$ is also $\mathbf\Delta^1_1$. It is the same observation that $\Sigma^1_1$ becomes $\mathbf\Sigma^1_1$.
Is it possible that there is some $A\in\Sigma^1_1$ is not $\Delta^1_1$, but is $\mathbf\Delta^1_1$? If this is true, is there any good characterization of this phenomenon? Can $\Sigma^1_1$ be replaced by $\Sigma^1_n$, or even $\Sigma^2_n$?