Nerves of generalized multicategories
Soichiro Fujii, Stephen Lack
TL;DR
The paper introduces $T$-simplicial objects and a nerve construction for $T$-categories, establishing a fully faithful embedding of $ extbf{Cat}_T(\mathcal{E})$ into $s_T(\mathcal{E})$ and a Segal-type condition characterizing the image. It develops a rich hierarchy of intermediate notions (T-graphs, magmoids, semicategories, reflexive/unital variants) and shows how they assemble into a nerve framework that generalizes internal categories in the Kleisli/monadic setting. The authors prove comonadicity results, endow the nerve category with simplicial enrichment, and lift these structures to a robust 2-category $ extbf{Cat}_T(\mathcal{E})$, including powers by arrows and local presentability under suitable hypotheses. The work unifies generalized multicategories, enrichment, and presentability in a coherent 2-categorical context, providing foundational tools for further study of $T$-categorical enrichment and presentable higher-categorical structures.
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.