T L  C S C B  C

In a computational process, certain entities (for example sets or arrays) and operations on them may be automatically available, for example by being provided by the programming language. We define background classes to formalize this idea, and we study some of their basic properties. The present notion of background class is more general than the one we introduced in an earlier paper, and it thereby corrects one of the examples in that paper. The greater generality requires a non-trivial notion of equivalence of background classes, which we explain and use. Roughly speaking, a background class assigns to each set (of atoms) a structure (for example of sets or arrays or combinations of these and similar entities), and it assigns to each embedding of one set of atoms into another a standard embedding between the associated background structures. We discuss several, frequently useful, properties that background classes may have, for example that each element of a background structure depends (in some sense) on only finitely many atoms, or that there are explicit operations by which all elements of background structures can be produced from atoms. ∗Mathematics Department, University of Michigan, Ann Arbor, MI 48109–1043, U.S.A., [email protected]