Dagger Categories of Tame Relations

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• 作者：Bart Jacobs (1)
  
• 关键词：Primary 68Q55 ; Secondary 18D10 ; 81P68 ; Dagger category ; quantum semantics
• 刊名：Logica Universalis
• 出版年：2013
• 出版时间：September 2013
• 年：2013
• 卷：7
• 期：3
• 页码：341-370
• 全文大小：402 KB
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• 作者单位：Bart Jacobs (1)
  
  1. Institute for Computing and Information Sciences (iCIS), Radboud University Nijmegen, Nijmegen, The Netherlands
  
• ISSN：1661-8300

文摘

Within the context of an involutive monoidal category the notion of a comparison relation ${\mathsf{cp} : \overline{X} \otimes X \rightarrow \Omega}$ is identified. Instances are equality = on sets, inequality ${\leq}$ on posets, orthogonality ${\perp}$ on orthomodular lattices, non-empty intersection on powersets, and inner product ${\langle {-}|{-} \rangle}$ on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with “tame-relations as morphisms. Examples include familiar categories in the foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation.