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arXiv · 1902.08848

Algebraic Type Theory and Universe Hierarchies

Jonathan Sterling
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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.

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Jonathan Sterling. 2019-02-23. Algebraic Type Theory and Universe Hierarchies. https://arxiv.org/abs/1902.08848

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