Degrees of Categoricity Above Limit Ordinals

A computable structure $mathcal{A}$ has degree of categoricity $mathbf{d}$ if $mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $mathcal{A}$. Fokina, Kalimullin, and Miller showed that every degree d.c.e. in and above $mathbf{0}^{(n)}$, for any $n < omega$, and also the degree $mathbf{0}^{(omega)}$, are degrees of categoricity. Later, Csima, Franklin, and Shore showed that every degree $mathbf{0}^{(alpha)}$ for any computable ordinal $alpha$, and every degree d.c.e. in and above $mathbf{0}^{(alpha)}$ for any successor ordinal $alpha$, is a degree of categoricity. We show that every degree c.e. in and above $mathbf{0}^{(alpha)}$, for $alpha$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e. in and above $mathbf{0}^{(omega)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk.