Homotopy theories of $(\infty, \infty)$-categories as universal fixed points with respect to enrichment

Abstract

We show that both the $\infty$-category of $(\infty, \infty)$-categories with inductively defined equivalences, and with coinductively defined equivalences, satisfy universal properties with respect to weak enrichment in the sense of Gepner and Haugseng. In particular, we prove that $(\infty, \infty)$-categories with coinductive equivalences form a terminal object in the $\infty$-category of fixed points for enrichment, and that $(\infty, \infty)$-categories with inductive equivalences form an initial object in the subcategory of locally presentable fixed points. To do so, we develop an analogue of Ad\'amek's construction of free endofunctor algebras in the $\infty$-categorical setting. We prove that $(\infty, \infty)$-categories with coinductive equivalences form a terminal coalgebra with respect to weak enrichment, and $(\infty, \infty)$-categories with inductive equivalences form an initial algebra with respect to weak enrichment.