Elementary equivalences and accessible functors

We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable categories. This notion specializes to L∞λ-equivalence and L∞λ-elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed colimits. We recover and extend results of Feferman and Eklof on “local functors” without fixing a language in advance. This is convenient for formalizing Lefschetz's principle in algebraic geometry, which was one of the main applications of the work of Eklof.