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 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 bireflective subcategory of A is itself a presheaf category. Given a small symmetric monoidal category C, we define diagonal structure on C, which is that structure and a little less than those axioms required to prove the monoidal structure is finite product structure. We then obtain a bireflective subcategory of [C ; Set] and deduce results relating its finite product structure with ; Set] determined by that of C. We also investigate closed structure