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The notion of a category is a concise concept shared among “groups and group homomorphisms”, “set and set-theoretic mappings”, “topological spaces and continuous mappings”, e t c.

Let be an abelian category. We will define the category Co() of (cochain) complexes as follows. An object in Co() is a sequence of objects and morphisms

A spectral sequence in an abelian category consists of doubly indexed objects of :

Let be an abelian category and let Co() be the category of bounded from below complexes as before.

The most fundamental object of study in Algebraic Geometry is the set of (or the number) of solutions of a system of (polynomial) equations.