Concept
Cubical type theory
Overview
In mathematical logic and theoretical computer science, cubical type theory is a flavor of type theory which gives a computational interpretation to univalent foundations (also known as homotopy type theory).
In cubical type theory, function extensionality and univalence are not postulated as axioms, but rather can be proved as theorems. Unlike traditional homotopy type theory, where the postulate of univalence creates stuck closed terms, cubical type theory has the canonicity property. It also enjoys normalization.
This is achieved by adding geometric primitives to the core rules of the type theory, including a formal interval object, interval variables, and operations for filling partial cubes.
The earliest cubical type theory is CCHM cubical type theory, named after its inventors Cohen, Coquand, Huber and Mörtberg. Later variants include Cartesian cubical type theory.
Cubical type theories have semantics in various types of cubical sets.
The Agda proof assistant includes an implementation of cubical type theory.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
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Then _R(A, B, C)_ is another eternal object of the lowest complex grade. Analogously there are eternal objects of successively higher grades. In respect to any complex eternal object, _S(D1, D2, ... Dn)_, the eternal objects _D1, ... Dn_, whose individual essences are constitutive of the individual essence of _S(D1, ... Dn)_, are called the components of _S(D1, ... Dn)_. It is obvious that the grade of complexity to…
Alfred North Whitehead, Science and the modern world
States what Cubical type theory is.
Hence, in the conclusion of a syllogism we restrict a predicate to a certain object, after having thought it in the major in its whole extent under a certain condition. This complete quantity of the extent in relation to such a condition is called universality (universalitas). To this corresponds totality (universitas) of conditions in the synthesis of intuitions.
Immanuel Kant, The Critique of Pure Reason
States what Cubical type theory is.
2º Once in possession of the concept of the mathematical continuum, is one safe from contradictions analogous to those which gave birth to it? No, and I will give an example.
Henri Poincaré, The Foundations of Science
States what Cubical type theory is.
Related ideas
Place in the tradition
Part of Type theory.