Degrees of bi-embeddable categoricity of equivalence structures

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) ${\mathrm{\Delta }}_{\alpha }^0$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of ${\mathrm{\Delta }}_{\alpha }^0$ bi-embeddable categoricity and relative ${\mathrm{\Delta }}_{\alpha }^0$ bi-embeddable categoricity coincide for equivalence structures for $\alpha =1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $\mathbf{0},{\mathbf{0}}^{\mathrm{\prime }}$, or ${\mathbf{0}}^{\mathrm{\prime }\mathrm{\prime }}$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability.