Geodesic
Familial 2-functors and parametric right adjoints
Theory and applications of categories, Tome 18 (2007), pp. 665-732.

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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 $\omega$-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, 1980], and those aspects of that theory that we require, that weren't discussed in [Weber, 2007], are reviewed here.
Classification : 18D50, 55P48
Keywords: nerves, parametric right adjoints, operads, familial 2-functors, fibrations, 2-categories 
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     author = {Mark Weber},
     title = {Familial 2-functors and parametric right adjoints},
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     volume = {18},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2007_18_a21/}
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Mark Weber. Familial 2-functors and parametric right adjoints. Theory and applications of categories, Tome 18 (2007), pp. 665-732. http://geodesic.mathdoc.fr/item/TAC_2007_18_a21/