Nerves of generalized multicategories

Abstract

For any category $E$ and monad T thereon, we introduce the notion of T-simplicial object in $E$. 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 ${\mathrm{Cat}}_T\left(E\right)$ of T-categories to the category $s_{T}E$ of T-simplicial objects, whose essential image is characterized by a simple condition. We show that the category $s_{T}E$ is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on ${\mathrm{Cat}}_T\left(E\right)$. We also study enriched limits and colimits in $s_{T}E$ and ${\mathrm{Cat}}_T\left(E\right)$, and show that if $E$ is locally finitely presentable and T is finitary, then ${\mathrm{Cat}}_T\left(E\right)$ is locally finitely presentable as a 2-category and $s_{T}E$ is locally finitely presentable as a simplicially-enriched category.