Abstract
AbstractWe introduce type-theoretic algebraic weak factorisation systems and show how they give rise to homotopy-theoretic models of Martin-Löf type theory. This is done by showing that the comprehension category associated with 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 and cubical sets models, as well as new models based on normal fibrations.
Highlights
Context and motivation The construction of category-theoretic models of MartinLof type theory [29] is a complex task that involves two main problems
One has to transform the category under consideration into a genuine model of Martin-Lof type theory, in which certain strictness conditions are required to hold, as in a split comprehension category [21, 22]
Our aim in this paper is to show that this impression is wrong and that the right adjoint splitting can be applied to obtain homotopy-theoretic models of Martin-Lof type theory
Summary
Context and motivation The construction of category-theoretic models of MartinLof type theory [29] is a complex task that involves two main problems. The main technical machinery used in the proof of Theorem 5.10 is Proposition 5.8, where we show that given a type-theoretic suitable topos, the resulting awfs of uniform fibrations can be equipped with a stable functorial choice of path objects, which is the structure necessary to produce pseudostable identity types. This result fills the gap between the theory developed in [14] and its intended application to the construction of models of Martin-Lof type theory.
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