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The category of implicative algebras and realizability

Published online by Cambridge University Press:  16 September 2019

Walter Ferrer Santos  and
Octavio Malherbe
Walter Ferrer Santos
Affiliation:
Facultad de Ciencias, Udelar, Iguá 4225, 11400 Montevideo, Uruguay Departamento de matemática, Centro Universitario de la región este, Udelar, Tacuarembó, 20100 Punta del Este, Uruguay, Email: wrferrer@cure.edu.uy
Octavio Malherbe*
Affiliation:
Departamento de matemática, Centro Universitario de la región este, Udelar, Tacuarembó, 20100 Punta del Este, Uruguay, Email: wrferrer@cure.edu.uy Facultad de Ingeniería, Udelar J. Herrera y Reissig 565, 11300 Montevideo, Uruguay
*
*Corresponding author. Email: malherbe@fing.edu.uy
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Abstract

In this paper, we continue with the algebraic study of Krivine’s realizability, completing and generalizing some of the authors’ previous constructions by introducing two categories with objects the abstract Krivine structures and the implicative algebras, respectively. These categories are related by an adjunction whose existence clarifies many aspects of the theory previously established. We also revisit, reinterpret, and generalize in categorical terms, some of the results of our previous work such as: the bullet construction, the equivalence of Krivine’s, Streicher’s, and bullet triposes and also the fact that these triposes can be obtained – up to equivalence – from implicative algebras or implicative ordered combinatory algebras.

Keywords

Realizabilityapplicative morphismscomputationally dense morphismscombinatory algebrasabstract Krivine’s structures
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 10 , November 2019 , pp. 1575 - 1606
Copyright
© Cambridge University Press 2019 

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