Algebraic Type Theory and Universe Hierarchies

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.