On a Generalization of Equilogical Spaces

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Abstract

We use the theory of triposes to prove that every (non-degenerate) locale H is the set of truth values of a complete and co-complete quasi-topos into which the category of topological spaces embeds and the topos of sheaves over H reflectively embeds.

Keywords

Triposes Partial equivalence relations Equilogical spaces 

Mathematics Subject Classification

Primary 03G30 Secondary 03B20 03B80 18B30 18C50 

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References

  1. 1.
    Bauer, F., Birkedal, L., Scott, D.S.: Equilogical spaces. Theor. Comput. Sci. 315(1), 35–59 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Scott, D.S.: A new category? Available at http://www.cs.cmu.edu/Groups/LTC/. Accessed 9 Apr 2018
  3. 3.
    Rosolini, G.: Equilogical spaces and filter spaces. Rend. Circ. Mat. Palermo 64, 157–175 (2000)MathSciNetMATHGoogle Scholar
  4. 4.
    Maietti, M.E., Pasquali, F., Rosolini, G.: Triposes, exact completions, and Hilbert’s \(\epsilon \)-operator. Tbil. Math. J. 10(3), 141–166 (2017)MathSciNetMATHGoogle Scholar
  5. 5.
    Frey, J.: Triposes, q-toposes and toposes. Ann. Pure Appl. Logic 166(2), 232–259 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Maietti, M.E., Rosolini, G.: Elementary quotient completion. Theor. Appl. Categ. 27, 445–463 (2013)MathSciNetMATHGoogle Scholar
  7. 7.
    Maietti, M.E., Rosolini, G.: Quotient completion for the foundation of constructive mathematics. Log. Univ. 7(3), 371–402 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Maietti, M.E., Rosolini, G.: Unifying exact completions. Appl. Categ. Struct. 23(1), 43–52 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Maietti, M.E., Rosolini, G.: Relating quotient completions via categorical logic. In: Probst, D., Schuster, P. (eds.) Concepts of Proof in Mathematics, Philosophy, and Computer Science, pp. 229–250. De Gruyter, Berlin (2016)Google Scholar
  10. 10.
    Pasquali, F.: A co-free construction for elementary doctrines. Appl. Categ. Struct. 23(1), 29–41 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Pasquali, F.: Remarks on the tripos to topos construction: comprehension, extensionality, quotients and functional-completeness. Appl. Categ. Struct. 24(2), 105–119 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hyland, J.M.E., Johnstone, P.T., Pitts, A.M.: Tripos theory. Math. Proc. Camb. Philos. Soc. 88, 205–232 (1980)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pitts, A.M.: Tripos theory in retrospect. Math. Struct. Comput. Sci. 12, 265–279 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    van Oosten, J.: Realizability: An Introduction to its Categorical Side, Volume 152 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co, North Holland (2008)Google Scholar
  15. 15.
    Carboni, A., Rosolini, G.: Locally Cartesian closed exact completions. Category theory and its applications (Montreal, QC, 1997). J. Pure Appl. Algebra 154(1–3), 103–116 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Johnstone, P.T.: Sketches of an Elephant—A Topos Theory Compendium. Clarendon Press, Oxford (2002)MATHGoogle Scholar
  17. 17.
    Pasquali, F.: A categorical interpretation of the intuitionistic, typed, first order logic with Hilbert’s \(\varepsilon \)-terms. Log. Univ. 10(4), 407–418 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pasquali, F.: Hilbert’s \(\varepsilon \)-operator in doctrines. IFCoLog J. Logics Their Appl. 4(2), 381–400 (2017)Google Scholar
  19. 19.
    Higgs, D.: Injectivity in the topos of complete Heyting algebra valued sets. Can. J. Math. 36(3), 550–568 (1984).  https://doi.org/10.4153/CJM-1984-034-4 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics Tullio Levi-CivitaUniversity of PadovaPaduaItaly

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