// Workers AI · dad joke modeWhat did the Freyd cover say? I've got you covered.

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

See also

• Artin gluing

References

• Freyd, P. J.; Scedrov, A. (22 November 1990). Categories, Allegories. Elsevier Science. ISBN 978-0-444-70368-2.
• Lambek, Joachim; Scott, Philip J. (1994). Introduction to higher order categorical logic (Paperback (with corr.), reprinted ed.). Cambridge: Cambridge Univ. Press. ISBN 9780521356534.
• Mitchell, John C.; Scedrov, Andre (1993). "Notes on sconing and relators". Computer Science Logic. Lecture Notes in Computer Science. Vol. 702. pp. 352–378. doi:10.1007/3-540-56992-8_21. ISBN 978-3-540-56992-3.
• Moerdijk, Ieke (1983). "On the Freyd cover of a topos". Notre Dame Journal of Formal Logic. 24 (4). doi:10.1305/ndjfl/1093870454. hdl:2066/128987.
• Scedrov, Andrej; Scott, Philip J. (1982). "A Note on the Friedman Slash and Freyd Covers". Studies in Logic and the Foundations of Mathematics. Vol. 110. pp. 443–452. doi:10.1016/S0049-237X(09)70142-9. ISBN 978-0-444-86494-9.
• Constructive Mathematics. Lecture Notes in Mathematics. Vol. 873. 1981. doi:10.1007/BFb0090721. ISBN 978-3-540-10850-4.

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. hdl:10044/1/95194.
• 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.