Cores of Countably Categorical Structures

A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism. We prove that every ω-categorical structure has a core. Moreover, every ω-categorical structure is homomorphically equivalent to a model-complete core, which is unique up to isomorphism, and which is finite or ω-categorical. We discuss consequences for constraint satisfaction with ω-categorical templates.