forming a very special case of regular categories. Examples As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any small abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem). If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over k. If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels. If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category. In this way, abelian categories show up in algebraic topology and algebraic geometry. If C is a small category and A is an abelian category, then the category of all functors from C to A forms an abelian category. If C is small and preadditive, then the category of all additive functors from C to A also forms an abelian category. The latter is a generalization of the R-module example, since a ring can be understood as a preadditive category with a single object. Grothendieck's axioms In his Tōhoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category A might satisfy. These axioms are still in common use to this day. They are the following: AB3) For every indexed family (Ai) of objects of A, the coproduct *Ai exists in A (i.e. A is cocomplete). AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism. AB5) A satisfies AB3), and filtered colimits of exact sequences are exact.