derived functors

Derived functors are mathematical tools used to extend the scope of simpler operations (functors) to capture more subtle information, especially when dealing with structures that are complicated or "twisted." When a functor doesn't preserve certain properties exactly, derived functors help measure and understand these deviations. Think of them as a way to systematically analyze what is lost or gained when transforming complex objects, providing deeper insights into their underlying structure. They are fundamental in areas like algebraic topology and homological algebra, offering a nuanced perspective on how mathematical objects relate to each other beyond initial straightforward mappings.