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.

43 pages, final journal version