TY - GEN A1 - Bodirsky, Manuel T1 - The core of a countably categorical structure N2 - A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for the classification for the computational complexity 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 this result remains valid for countably categorical structures, and prove that every countably categorical structure has a core, which is unique up to isomorphism, and which is again countably categorical. We thus reduced the classification for the complexity of constraint satisfaction problems with countably categorical templates to the classifiaction for constraint satisfaction problems where the templates are countably categorical cores. We also show that a core of a countably categorical structure Gamma is model complete, and therefore universal-existential axiomatizable. If Gamma contains all primitive positive definable relations, then the core of Gamma admits quantifier elimination. We discuss consequences for constraint satisfaction with countably categorical templates. Y1 - 2005 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/275 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-2754 ER -