Liftings of functors (Q1066991)

The author generalizes the work of, among others, \textit{M. Barr} [Math. Z. 116, 307-322 (1970; Zbl 0194.017)]. One of two dual principal results is: Let \({\mathbb{T}}=(T,\eta,\mu)\) be a triple in the category K; and \({\mathbb{C}}=(C,\epsilon,\delta)\), a cotriple in, L. A morphism from \((K,(T,\eta,\mu))\) to \((L,(C,\epsilon,\delta))\) is a pair \((X,\lambda)\) where \(X: K\to L\) is a functor and \(\lambda: CXT\to X\) is a natural transformation satisfying \(\lambda \cdot CX\eta =\epsilon X\) and \(\lambda \cdot CX\mu =\lambda \cdot C\lambda T\cdot \delta XT^ 2\). Each morphism \((X,\lambda)\) induces a functor \(\bar X: K_{{\mathbb{T}}}\to L_{{\mathbb{C}}}\), a lifting of X to the Kleisli categories, such that \(\bar XF_{{\mathbb{T}}}=F_{{\mathbb{C}}}X\). Inversely, such a functor V induces a natural transformation \(\lambda: CXT\to X\) defined by \(\lambda_ A=V(1_{TA})\) such that \((X,\lambda)\) is a morphism. Further, both steps are mutually inverse bijections.