# Degrees of bi-embeddable categoricity

## TL;DR

This paper explores the complexity of embeddings between bi-embeddable structures, extending categoricity spectrum concepts, identifying structures without a degree of bi-embeddable categoricity, and characterizing degrees that serve as such categoricity degrees.

## Contribution

It extends known results on categoricity spectra to bi-embeddability, introduces the bi-embeddable categoricity spectrum, and characterizes degrees that can serve as degrees of bi-embeddable categoricity.

## Key findings

- Structures without degree of bi-embeddable categoricity identified.
- Every d.c.e. degree above certain levels is a degree of bi-embeddable categoricity.
- Examples of degrees not serving as bi-embeddable categoricity spectra provided.

## Abstract

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $\mathcal A$ as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of $\mathcal A$; the degree of bi-embeddable categoricity of $\mathcal A$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $\mathbf{0}^{(\alpha)}$ for $\alpha$ a computable successor ordinal and $\mathbf{0}^{(\lambda)}$ for $\lambda$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.