Type theory | Abstract algebra | Category theory | Proof theory

Setoid

In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set). (Wikipedia).

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Erik Palmgren: From type theory to setoids and back

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Setoids, e-Categories, and Exact Completions - Richard Garner

Richard Garner Queen Mary, University of London March 7, 2013 For more videos, visit http://video.ias.edu

From playlist Mathematics

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Fabio Pasquali: Assemblies as an elementary quotient completion

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From playlist Workshop: "Types, Homotopy, Type theory, and Verification"

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Jacopo Emmenegger: W types in the setoid model

The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstact: Current arguments to obtain initial algebras for polynomial endofunctors in categories of equivalence relations rely on assumptions like UIP or on constructions that invol

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On the Setoid Model of Type Theory - Erik Palmgren

Erik Palmgren University of Stockholm October 18, 2012 For more videos, visit http://video.ias.edu

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Related pages

Extension (semantics) | Intuitionistic type theory | Equality (mathematics) | Quotient set | Groupoid | Quotient type | Rational number | Foundations of mathematics | Proof theory | Coq | Errett Bishop | Curry–Howard correspondence | Real analysis | Mathematics | Partial equivalence relation | Set (mathematics) | Real number | Apartness relation | Equivalence relation | Algorithm