Some Fundamental Algebraic Tools for the Semantics of Computation - Part III: Indexed Categories
A Tarlecki, R M Burstall and J A GoguenAbstract: We present the concept of indexed category, a technical tool to model families of categories defined in a uniform way. We show how any indexed category gives rise to a single flat category, a disjoint union of the components with some additional morphisms between them. Similarly, any indexed functor (a family of functors between component categories) induces a flat functor between the corresponding flat categories. We prove that under some technical conditions flat categories are complete (resp. cocomplete) if all their components are so; flat functors have left adjoints if all their components do. A few examples illustrate the usefulness of these concepts and results.
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