Nerves of generalized multicategories

Abstract

For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction defines a fully faithful functor from the category $\mathbf{Cat}_T({\mathcal E})$ of $T$-categories to the category $s_T({\mathcal E})$ of $T$-simplicial objects, whose essential image is characterized by a simple condition. We show that the category $s_T({\mathcal E})$ is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on $\mathbf{Cat}_T({\mathcal E})$. We also study enriched limits and colimits in $s_T({\mathcal E})$ and $\mathbf{Cat}_T({\mathcal E})$, and show that if ${\mathcal E}$ is locally finitely presentable and $T$ is finitary, then $\mathbf{Cat}_T({\mathcal E})$ is locally finitely presentable as a 2-category and $s_T({\mathcal E})$ is locally finitely presentable as a simplicially-enriched category.