Bireflectivity

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity, Bireflective subcategories o f a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of splitidempotent natural transformations on id(A). In the special case that,A is a presheaf category, we characterise them in terms of the domain, and prove that any bireflective subcategory of A is itself a presheaf category. We define diag onal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obta in a bireflective subcategory of [C-op, Set] and deduce results relating it s finite product structure with the monoidal structure of [C-op, Set] deter mined by that of C. We also investigate the closed structure, Finally, for completeness, we give results on bireflective subcategories in Rel(A), the category of relations in a topos A: and a characterisation of bireflection functors in terms of modules they define. (C) 1999 Elsevier Science B.V, AI I rights reserved.