Type theory | Abstract algebra | Category theory | Proof theory
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).
Erik Palmgren: From type theory to setoids and back
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstract: Errett Bishop proposed several type-theoretic languages to be used for formalization of constructive mathematics. The notion of set of (Bishop-Bridges 1985) always requi
From playlist Workshop: "Constructive Mathematics"
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
Fabio Pasquali: Assemblies as an elementary quotient completion
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstarct: We characterize the category of assemblies as an elementary quotient completion, a construction introduced by Maietti and Rosolini to axiomatize the category of setoids u
From playlist Workshop: "Types, Homotopy, Type theory, and Verification"
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
From playlist Workshop: "Types, Homotopy, Type theory, and Verification"
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
From playlist Mathematics