On the localness of the embedding of algebras

Abstract

Let N : A → B be a faithful functor between categories. Given an object B of B, we may ask whether there is an embedding B → NA with A ∈ A. In some cases the answer is well known. For instance, an abelian semigroup may be embedded in an abelian group if and only if it is cancellative. And every Lie algebra over a field K is embeddable in an associative K-algebra with identity. Many other examples are known. This paper concentrates on the localness of the embeddability. That is, it studies conditions under which the following property holds: B ∈ B is embeddable in NA for some object A of A, whenever every finitely generated subobject of B is so.