derivedfunktorien

Derivedfunktorien is a term used in abstract algebra, specifically within category theory, to describe a way of extending functors to operate on derived categories. A functor is a mapping between categories that preserves their structure. Derived categories are a more sophisticated construction than standard categories, often used to handle algebraic objects that may not be "well-behaved" in the original category. The process of derivation involves adapting a functor to work with the chain complexes and quasi-isomorphisms that are central to derived categories.

When a functor is derived, it generally produces a new functor that maps between the derived categories. This derived functor often captures essential information about the original functor's behavior, particularly its relationship to homology and cohomology. For example, if the original functor is additive, its derived version will be an endofunctor on the derived category of modules. The construction of derived functors is typically achieved through a process called " विकरण" (resolution) or by using projective or injective objects. This allows for the study of phenomena that are obscured in the original category, such as Ext and Tor functors. The concept of derived functors is fundamental in fields like algebraic topology, algebraic geometry, and homological algebra.