Logica Universalis

, Volume 12, Issue 1–2, pp 129–140 | Cite as

On a Generalization of Equilogical Spaces

  • Fabio Pasquali


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.


Triposes Partial equivalence relations Equilogical spaces 

Mathematics Subject Classification

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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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 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). 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

Personalised recommendations