Bireflectivity

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireeectivity. Bireeective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of split-idempotent natural transformations on id A. In the special case that A is a presheaf category, we characterize them in terms of the domain, and prove that any bireeective subcategory of A is itself a presheaf category. Given a small symmetric monoidal category C, we deene diagonal structure on C, which is that structure and a little less than those axioms required to prove the monoidal structure is nite product structure. We then obtain a bireeective subcategory of C op ; Set] and deduce results relating its nite product structure with the monoidal structure of C op ; Set] determined by that of C. We also investigate closed structure.