Bireflectivity

P. J. Freyd, P. W. O'Hearn, A. J. Power, M. Takeyama*, R. Street, R. D. Tennent

*Corresponding author for this work

    Research output: Contribution to journalArticle

    5 Citations (Scopus)

    Abstract

    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 idscript 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 signop, Set] and deduce results relating its finite product structure with the monoidal structure of [script C signop, 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.

    Original languageEnglish
    Pages (from-to)49-76
    Number of pages28
    JournalTheoretical Computer Science
    Volume228
    Issue number1-2
    Publication statusPublished - 28 Oct 1999

    Keywords

    • Categories of relations
    • Modules
    • Monoidal structure
    • Presheaves
    • Reflective subcategories
    • Split-idempotents

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