Nerves of generalized multicategories

TL;DR

This paper develops a nerve construction for T-categories in a category E, establishing a fully faithful functor to T-simplicial objects, and explores their enriched limits, colimits, and presentability properties.

Contribution

It introduces T-simplicial objects and a nerve functor for T-categories, characterizing their essential image and analyzing their enriched categorical structures.

Findings

The nerve functor is fully faithful from T-categories to T-simplicial objects.

The category of T-simplicial objects is enriched over simplicial sets.

Under certain conditions, T-categories and T-simplicial objects are locally finitely presentable.

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.