Models of Martin-L\"of type theory from algebraic weak factorisation systems

Abstract

We introduce type-theoretic algebraic weak factorisation systems and show how they give rise to homotopy-theoretic models of Martin-L\"of type theory. This is done by showing that the comprehension category associated to a type-theoretic algebraic weak factorisation system satisfies the assumptions necessary to apply a right adjoint method for splitting comprehension categories. We then provide methods for constructing several examples of type-theoretic algebraic weak factorisation systems, encompassing the existing groupoid model and cubical sets models, as well as some models based on normal fibrationsComment: Changed title (it used to be "Type-theoretic algebraic weak factorisation systems"); rewritten introduction; fixed typos; fixed inaccuracy in Lemma 2.3 spotted by Paige North; added references. 37 page