Extensions and homological algebra

If you ever followed a course in algebra, you may have heard about extensions of “things”. Most “things” fit in what is called a category, that is, an abstract structure remembering how to compose morphisms between these “things”, and sometimes what is the sum of such “things”, the kernel of a morphism: we are interested in the structure of abelian category, which is the framework of vector spaces, modules over a ring, sheaves of modules… An extension of A by B is an exact sequence

where is the kernel, and is the quotient (cokernel). The group of extensions is the set of isomorphism classes of such exact sequences.