Nerves of bicategories as stratified simplicial sets (Q1008742)

The paper builds on the work of \textit{R. Street} [Contemp. Math. 318, 207--213 (2003; Zbl 1039.18005)] who proposed to define a week \(\omega\)-category as a weak complicial set, i.e., a stratified simplicial set satisfying some additional conditions. The author modifies this idea to give a definition (actually two different definitions) of a weak \(n\)-category. He also shows that the resulting category of weak \(n\)-categories is equivalent, for \(n=1\), to the category of small categories and, for \(n=2\), to the category of bicategories. This demonstrates that at least in these low dimensions the proposed definition behaves as expected. The proof of equivalence between week \(2\)-categories and bicategories relies on the characterization of nerves of bicategories developed by \textit{J. W. Duskin} [Theory Appl. Categ. 9, 198--308 (2001; Zbl 1046.18009)].