Modularity and descent Original Research Article

We show that a left exact category image 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 image is monadic and the corresponding monad T is nullary, i.e. the canonical maps T(0) + X → T(X) are isomorphisms, and the adjoint is comonadic (and then also the corresponding comonad is conullary). This result should be compared with the well-known characterization of additive categories as those for which the same forgetful functor is an equivalence.