A derived category is a mathematical concept used in the field of homological algebra. It provides a way to study complexes of objects, such as modules or sheaves, by focusing on their homological properties. Derived categories allow mathematicians to work with morphisms and cohomology in a more flexible manner, enabling the analysis of relationships between different algebraic structures. In a derived category, objects are typically complexes of abelian groups or vector spaces, and morphisms are defined up to homotopy. This framework helps in understanding various mathematical phenomena, including derived functors and triangulated categories, which are essential for modern algebraic geometry and representation theory.