Nerves of generalized multicategories

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 Cat_T(E) of T -categories to the category s_TE of T -simplicial objects, whose essential image is characterized by a simple condition. We show that the category s_TE is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on Cat_T(E). We also study enriched limits and colimits in s_TE and Cat_T(E), and show that if E is locally finitely presentable and T is finitary, then Cat_T(E) is locally finitely presentable as a 2-category and s_TE is locally finitely presentable as a simplicially-enriched category.