Familial 2-functors and Parametric Right Adjoints

We define and study familial 2-functors primarily with a view to the development of the 2-categorical approach to operads of [Weber, 2005]. Also included in this paper is a result in which the well-known characterisation of a category as a simplicial set via the Segal condition, is generalised to a result about nice monads on cocomplete categories. Instances of this general result can be found in [Leinster, 2004], [Berger, 2002] and [Moerdijk-Weiss, 2007b]. Aspects of this general theory are then used to show that the composite 2-monads of [Weber, 2005] that describe symmetric and braided analogues of the ω-operads of [Batanin, 1998], are cartesian 2-monads and their underlying endo-2-functor is familial. Intricately linked to the notion of familial 2-functor is the theory of fibrations in a finitely complete 2-category [Street, 1974] [Street, 1980a], and those aspects of that theory that we require, that weren’t discussed in [Weber, 2007], are reviewed here.