Higher categorical structures are often defined by induction on dimension, which a priori produces only finite-dimensional structures. In this paper we show how to extend such definitions to infinite dimensions using the theory of terminal coalgebras, and we apply this method to Trimble's notion of weak n-category. Trimble's definition makes explicit the relationship between n-categories and topological spaces; our extended theory produces a definition of Trimble $\infty$-category and a fundamental $\infty$-groupoid construction.
Furthermore, terminal coalgebras are often constructed as limits of a certain type. We prove that the theory of Batanin - Leinster weak $\infty$-categories arises as just such a limit, justifying our approach to Trimble $\infty$-categories. In fact we work at the level of monads for $\infty$-categories, rather than $\infty$-categories themselves; this requires more sophisticated technology but also provides a more complete theory of the structures in question.
Keywords: $\infty$-category, $\omega$-category, n-category, higher category, terminal coalgebra, final coalgebra
2010 MSC: 18D05 (primary), 18C15
Theory and Applications of Categories, Vol. 34, 2019, No. 34, pp 1073-1133.