Twisted arrow $\infty$-categories of $(\infty ,1)$-categories were introduced by Lurie, and they have various applications in higher category theory. Abellán García and Stern gave a generalization to twisted arrow $\infty$-categories of $(\infty ,2)$-categories. In this paper, we introduce another simple model for twisted arrow $\infty$-categories of $(\infty ,2)$-categories.

In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete group G. We show that in two of the models we can also consider actions of any simplicial group G.

We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category Θn. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk Θn-spaces, showing that n-quasi-categories are a model for (∞, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.

We introduce a homotopy 2-category structure on the collection of dg-categories, dg-functors, and their derived transformations. This construction provides for a conceptual proof of Deligne's conjecture on Hochschild cochains.

The usual constructions of classifying spaces for monoidal categories produce $\text{CW}$ -complexes with many cells that,moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.