Multitensors as monads on categories of enriched graphs

In this paper we unify the developments of [Batanin, 1998], [Batanin- Weber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented here is the conceptual background for subsequent work: in [Batanin-Cisinski-Weber, 2012] the Gray tensor product of 2-categories and the Crans tensor product [Crans, 1999] of Gray categories are exhibited as existing within our framework, and in [Weber, 2013] the explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads.