Burroni's \( T \)-categories generalize internal categories, which themselves generalize ordinary (small) categories. The classical nerve construction assigns to a small category a simplicial set, and this construction extends routinely to internal categories: any internal category in a category \( \mathcal E \) gives rise to a simplicial object in \( \mathcal E \) as its nerve. In recent joint work with Steve Lack, titled Nerves of generalized multi-categories, we extend the nerve construction further to Burroni's \( T \)-categories. For a category \( \mathcal E \) equipped with a monad \( T \), we introduce the notion of a \( T \)-simplicial object in \( \mathcal E \) and show that every \( T \)-category determines a \( T \)-simplicial object. Moreover, those \( T \)-simplicial objects arising from \( T \)-categories admit a simple characterization. In this talk, I will explain these results and discuss an application to the study of powerful (or exponentiable) \( T \)-functors.
This talk is based on ongoing joint work with Steve Lack.