Algebraic Type Theory and Universe Hierarchies (1902.08848v1)

Abstract: It is commonly believed that algebraic notions of type theory support only universes `a la Tarski, and that universes `a la Russell must be removed by elaboration. We clarify the state of affairs, recalling the details of Cartmell's discipline of generalized algebraic theory, showing how to formulate an algebraic version of Coquand's cumulative cwfs with universes `a la Russell. To demonstrate the power of algebraic techniques, we sketch a purely algebraic proof of canonicity for Martin-L\"of Type Theory with universes, dependent function types, and a base type with two constants.