Abstract
We show that the classifying category C ( T ) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.
Highlights
From the point of view of mathematical logic and theoretical computer science, Martin-Lof’s axioms for identity types [24] admit a conceptually clear explanation in terms of the propositions-as-types paradigm [14, 21, 27]
The difficulty of isolating the structure determined by identity types is closely related to the problem of describing a satisfactory category-theoretic semantics for them
One approach to obtain a semantics of identity types that does not validate the reflection rules is to consider categories equipped with a weak factorisation system [2]
Summary
From the point of view of mathematical logic and theoretical computer science, Martin-Lof’s axioms for identity types [24] admit a conceptually clear explanation in terms of the propositions-as-types paradigm [14, 21, 27]. GARNER with the axioms for identity types, its classifying category C(T) admits a non-trivial weak factorisation system, which we shall refer to as the identity type weak factorisation system This result should be regarded as analogous to the fundamental result exhibiting the structure of a cartesian closed category on the classifying category of the -typed lambda calculus [19, 28]. After having established the existence of the identity type weak factorisation system, we will provide an explicit characterisation of its classes of maps. This will lead us to two applications. We use the functor F to relate the identity type weak factorisation system to the natural Quillen model structure on the category of groupoids [1, 17], by showing how the identity type weak factorisation system is mapped into the weak factorisation system determined by injective equivalences and Grothendieck fibrations in Gpd
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