Bireflectivity

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category script A sign are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of split-idempotent natural transformations on id_{script A sign}. In the special case that script A sign is a presheaf category, we characterise them in terms of the domain, and prove that any bireflective subcategory of script A sign is itself a presheaf category. We define diagonal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obtain a bireflective subcategory of [script C sign^op, Set] and deduce results relating its finite product structure with the monoidal structure of [script C sign^op, Set] determined by that of script C sign. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(script A sign), the category of relations in a topos script A sign, and a characterisation of bireflection functors in terms of modules they define.