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We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bifunctor with respect to the bicategorical model category of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings. AcknowledgementsThe first author is supported by GA ČR EXPRO 19-28628X. The third author gratefully acknowledges the support of Praemium Academiae of M. Markl and RVO:67985840, as well as fruitful conversations with Nick Rozenblyum during his stay at MSRI.

The goal of this paper is to provide the last equivalence needed in order to identify all known models for (∞, 2)categories. We do this by showing that Verity's model of saturated 2-trivial complicial sets is equivalent to Lurie's model of ∞-bicategories, which, in turn, has been shown to be equivalent to all other known models for (∞, 2)categories. A key technical input is given by identifying the notion of ∞-bicategories with that of weak ∞-bicategories, a step which allows us to understand Lurie's model structure in terms of Cisinski-Olschok's theory. Several of our arguments use tools coming from a new theory of outer (co)-Cartesian fibrations, further developed in a companion paper. In the last part of the paper, we construct a homotopically fully faithful scaled simplicial nerve functor for 2-categories, give two equivalent descriptions of it, and show that the homotopy 2category of an ∞-bicategory retains enough information to detect thin 2-simplices.

In this paper we show that both the homotopy category of strict n-categories, 1 n ∞, and the homotopy category of Steiner's augmented directed complexes are equivalent to the category of homotopy types. In order to do so, we first prove a general result, based on a strategy of Fritsch and Latch, giving sufficient conditions for a nerve functor with values in simplicial sets to induce an equivalence at the level of homotopy categories. We then apply this result to strict n-categories and augmented directed complexes, where the assumption of our theorem were first established by Ara and Maltsiniotis and of which we give an independent proof.

We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an ω-category that is suitably free and loop-free. An instance of the result recovers the fact that the standard m-simplex is equivalent to the complicial nerve of the m-oriental.1 There are two other types of cone constructions (see [AM20, Rmk 6.37] and [GHL20, §4.2] for more details on how to construct them in ω-categories and in scaled simplicial sets), that would detect lax limits and oplax colimits, but they are not as easy to construct and the techniques from this paper cannot be employed on the nose to cover those cases.Acknowledgements. The first-named author gratefully acknowledges the support of Praemium Academiae of M. Markl and RVO:67985840.