We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not only for parametricity, but also for similar interpretations of type theory and in fact similar interpretations of any generalized algebraic theory. To be precise we consider a functor forgetting unary operations and equations defining them recursively in a generalized algebraic theory. We show that it has a right adjoint. We use techniques from locally presentable category theory, as well as from quotient inductive-inductive types.

翻译：我们构建了一种类型理论的模型,从任意的理论中可以得出相似的参数。新模型中的一种类型在旧模型中是一种半孵化型,可以说明参数和立方体之间的对应性。我们的构造不仅用于参数性,而且用于对类型理论的类似解释,事实上也用于对任何通用代数理论的类似解释。精确地说,我们把一种杀菌剂视为一种在普遍代数理论中逐渐忘记非遗传操作的方程式和对之进行交替定义的方程式。我们证明它具有一种正确的连接性。我们使用了当地现成的分类理论的技术,以及引导学的原始类型。