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    Kinoshita, Yoshiki and Power, John 2014. Category theoretic structure of setoids. Theoretical Computer Science, Vol. 546, p. 145.

  • Mathematical Structures in Computer Science, Volume 20, Issue 4
  • August 2010, pp. 563-576

Setoids and universes

  • DOI:
  • Published online: 07 April 2010

Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with universes that are as small as possible (and thus, the type theory is as weak as possible). In particular, we will consider epimorphisms and disjoint sums. We show that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably non-derivable.

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A. Carboni (1995) Some free constructions in realizability and proof theory. J. Pure Appl. Algebra 103 (2) 117148.

A. Carboni , S. Lack and R. F. C. Walters (1993) Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84 (2) 145158.

M. E. Maietti (2009) A minimalist two-level foundation for constructive mathematics. Ann. Pure Appl. Logic 160 (3) 319354.

R. Mines , F. Richman and W. Ruitenburg (1988) A course in constructive algebra, Universitext, Springer-Verlag.

J. M. Smith (1989) Propositional functions and families of types. Notre Dame J. Formal Logic 30 (3) 442458.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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