The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street's theory of fibrations within 2-categories is explored. Johnstone's notion of "bagdomain data" is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads.
Keywords: polynomial functors, 2-monads
2010 MSC: 18A05; 18D20; 18D50
Theory and Applications of Categories, Vol. 30, 2015, No. 16, pp 533-598.