Abstract

. We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd a m a dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZ-doctrine. 1. Introduction Free co-completions of categories under suitable classes of colimits were the motivating examples for the definition of KZ-doctrines. We introduce in this paper a not-strict version of such doctrines defined via a fully faithful adjoint string. Thus, a non-strict KZ-doctrine on a 2-category K consists of a normal endo homomorphism D : K \Gamma! K, and strong transformations d : 1K \Gamma! D, and m : DD \Gamma! D in such a way that Dd a m a dD forms a fully faithful adjoint string, satisfying one equation involving the unit of...

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