Canonical bases in excellent classes

Abstract

We show that any (atomic) excellent class 𝔎 can be expanded with hyperimaginaries to form an (atomic) excellent class 𝔎^eq which has canonical bases. When 𝔎 is, in addition, of finite U-rank, then 𝔎^eq is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class 𝔎 gives rise to an excellent class, which is simple if 𝔎 is of finite U-rank.