We study the functor ℓ2 from the category of partial injections to the category of Hilbert spaces. The former category is finitely accessible, and in both categories homsets are algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces.


Hilbert Space Orthonormal Basis Inverse Semigroup Polar Decomposition Partial Isometry 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Chris Heunen
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUK

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