We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets.To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model.As a corollary, we conclude that Martin-Löf type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

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We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for (n-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for (n-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler.Proof. We will apply [Ols09, Theorem 2.2.5] (with L taken to be all monomorphisms). Our functorial cylinder is given byNote that (1), ( 2) and (4) imply that X ⊗ (∂ 0 , ∂ 1 ) is a monomorphism for each X since it can be written as i X ⊗(∂ 0 , ∂ 1 ) where i X ∶ 0 → X is the unique map. Similarly, since any map from a terminal object (and in particular ∂ 0 , ∂ 1 ) is a monomorphism, we can deduce that X ⊗ ∂ 0 and X ⊗ ∂ 1 are always

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit an elegant and canonical notion of a mapping space between two objects.

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Löf Type Theory with dependent sums and intensional identity types is the internal language of (∞, 1)-categories with finite limits.