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We introduce type-theoretic algebraic weak factorisation systems and show howthey give rise to homotopy-theoretic models of Martin-L\"of type theory. Thisis done by showing that the comprehension category associated to atype-theoretic algebraic weak factorisation system satisfies the assumptionsnecessary to apply a right adjoint method for splitting comprehensioncategories. We then provide methods for constructing several examples oftype-theoretic algebraic weak factorisation systems, encompassing the existinggroupoid model and cubical sets models, as well as some models based on normalfibrations

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