Bireflectivity

Peter J. Freyd, Peter W. O'Hearn, A. John Power, Makoto Takeyama, R. Street, Robert D. Tennent

Abstract: Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A<math><mtext>A</mtext></math> are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of split-idempotent natural transformations on idA<math><mtext>id</mtext><msub><mi></mi><mn><mtext>A</mtext></mn></msub></math>. In the special case that A<math><mtext>A</mtext></math> is a presheaf category, we characterise them in terms of the domain, and prove that any bireflective subcategory of A<math><mtext>A</mtext></math> 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 [Cop,Set]<math><mtext>[</mtext><mtext>C</mtext><msup><mi></mi><mn><mtext>op</mtext></mn></msup><mtext>,</mtext><mtext>Set</mtext><mtext>]</mtext></math> and deduce results relating its finite product structure with the monoidal structure of [Cop,Set]<math><mtext>[</mtext><mtext>C</mtext><msup><mi></mi><mn><mtext>op</mtext></mn></msup><mtext>,</mtext><mtext>Set</mtext><mtext>]</mtext></math> determined by that of C<math><mtext>C</mtext></math>. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(A)<math><mtext>Rel</mtext><mtext>(</mtext><mtext>A</mtext><mtext>)</mtext></math>, the category of relations in a topos A<math><mtext>A</mtext></math>, and a characterisation of bireflection functors in terms of modules they define.