MODULARITY AND DESCENT

We show that a left exact category 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 from the category of abelian groups in E is m onadic and the corresponding monad T is nullary, i.e. the canonical ma ps T(0) + X --> T(X) are isomorphisms, and the adjoint is comonadic (a nd then also the corresponding comonad is conullary). This result shou ld be compared with the well-known characterization of additive catego ries as those for which the same forgetful functor is an equivalence.