Bire ectivity

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bire ectivity. Bire ective subcategories of 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 idA. In the special case that A is a presheaf category, we characterise them in terms of the domain, and prove that any bire ective subcategory of A is itself a presheaf category. We de ne 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 bire ective subcategory of [C;Set] and deduce results relating its nite product structure with the monoidal structure of [C;Set] determined by that of C. We also investigate the closed structure. Finally, for completeness, we give results on bire ective subcategories in Rel(A), the category of relations in a topos A, and a characterisation of bire ection functors in terms of modules they de ne. c © 1999 Elsevier Science B.V. All rights reserved.