Modularity and descent (Q1906926)

By studying the relations between the notion of effective descent morphism and the categorical version of the condition of modularity for lattices, the authors give a characterization of affine categories: a left exact category \({\mathcal E}\) with finite coproducts is affine, i.e., it is a slice of an additive category with kernels, if and only if the forgetful functor \(U : Ab ({\mathcal E}) \to {\mathcal E}\) is monadic and the corresponding monad is nullary, and the adjoint of \(U\) is comonadic. This characterization theorem only in terms of the functor \(U\) should be compared with the well-known characterization of additive categories (with kernels) as those for which the functor \(U\) is an equivalence.