In this talk, I will start by presenting the notion of complete Segal spaces, due to Rezk. These model (∞,1)-categories. Then we will see two ways of extending this notion into models of (∞,2)-categories: 2-fold complete Segal spaces (due to Barwick) and Theta-2-spaces (due to Rezk). Finally, we will see that the two models are equivalent.
Abstract: 2-quasi-categories are a model for (∞,2)-categories defined by Ara as the fibrant objects of a certain model structure on the category of $\Theta_2$-sets. In the first half of this talk, I will recall Ara's definition of 2-quasi-categories and Leinster's nerve construction, which embeds bicategories into 2-quasi-categories. In the second half, I will survey various constructions and results on 2-quasi-categories, including comparisons of 2-quasi-categories with Rezk's $\Theta_2$-spaces and with quasi-category-enriched Segal categories.
- Dimitri Ara. Higher quasi-categories vs higher Rezk spaces. J. K-Theory 14 (2014), no. 3, 701--749.
- Alexander Campbell. A homotopy coherent cellular nerve for bicategories. Preprint, arXiv:1907.01999 (2019).
- Yuki Maehara. Inner horns for 2-quasi-categories. Adv. Math. 363 (2020).