Bireflectivity

Abstract

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A 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 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 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 [ C op , Set ] and deduce results relating its finite product structure with the monoidal structure of [ C op , Set ] determined 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.