Familial 2-functors and parametric right adjoints

Theory and Applications of Categories, 18:665-732, 2007

We define and study familial 2-functors primarily with a view to the development of the 2-categorical approach to operads of [7]. 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 [3], [2] and [4]. Aspects of this general theory are then used to show that the composite 2-monads of [7] that describe symmetric and braided analogues of the omega-operads of Batanin [1] 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 [5] [6], and those aspects of that theory that we require, that weren't discussed in [8], are reviewed here.

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