Limitwise monotonic spectra of $\Sigma^0_2$-sets

In this paper, we study classes of Turing degrees, contained in limitwise monotonic spectra. In particular, we prove that limitwise monotonic spectra of $\Sigma^0_2$-sets are co-meager. Moreover, we construct a $\Sigma^0_2$-set whose limitwise monotonic spectrum is co-null.