• Source: Freyd cover

      • In the mathematical discipline of category theory, the Freyd cover or scone category is a construction that yields a set-like construction out of a given category. The only requirement is that the original category has a terminal object. The scone category inherits almost any categorical construct the original category has. Scones can be used to generally describe proofs that use logical relations.
      The Freyd cover is named after Peter Freyd. The other name, "scone", is intended to suggest that it is like a cone, but with the Sierpiński space in place of the unit interval.


      Definition


      Formally, the scone of a category C with a terminal object 1 is the comma category




      1

      Set


      ↓

      Hom

      C


      ⁡
      (
      1
      ,
      −
      )


      {\displaystyle 1_{\text{Set}}\downarrow \operatorname {Hom} _{C}(1,-)}

      .


      See also


      Artin gluing


      Notes




      References




      Further reading


      Johnstone, P. T. (1992). "Partial products, bagdomains and hyperlocal toposes §.6, Bagdomains and Scones". Applications of Categories in Computer Science. pp. 315–339. doi:10.1017/CBO9780511525902.018. ISBN 978-0-521-42726-5.
      Vickers, Steven (1999). "Topical categories of domains". Mathematical Structures in Computer Science. 9 (5): 569–616. doi:10.1017/S0960129599002741.
      Sterling, Jonathan; Harper, Robert (2021). "Logical Relations as Types: Proof-Relevant Parametricity for Program Modules". Journal of the ACM. 68 (6): 1–47. arXiv:2010.08599. doi:10.1145/3474834.

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