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

Dimitri Ara's 2-quasi-categories, which are certain presheaves over André Joyal's 2-cell category Θ 2 , are an example of a concrete model that realises the abstract notion of (∞, 2)-category. In this paper, we prove that the 2-quasi-categories and the fibrations into them can be characterised using the inner horn inclusions and the equivalence extensions introduced by David Oury. These maps are more tractable than the maps that Ara originally used and therefore our result can serve as a combinatorial foundation for the study of 2-quasi-categories.

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