Tendonderived

Tendonderived is a term used in contemporary category theory and homological algebra to describe a family of constructions that generalize derived functors by adding a supplementary structure called the tendon. Informally, the tendon encodes how information is transmitted along a chosen network of morphisms, allowing one to track homological data not only along the original functor but also through the tendon-compatibility.

Origin and definitions: The term tendonderived was proposed in discussions around augmented derived theories in the 2010s, with a formal treatment appearing in a hypothetical study by Rivera and Chen. The framework starts with an exact or abelian category A equipped with a tendon endofunctor T: A -> A together with natural transformations that express a compatibility condition with short exact sequences or complexes. For a functor F: A -> B that preserves the relevant structure, one defines tendonderived functors L^T F and R^T F as the left and right derived functors relative to T. The associated tendonderived category is obtained by localizing the homotopy category of chain complexes at a class of T-acyclic objects.

Relationship to classical theory: If T is the identity, tendonderived coincides with the ordinary derived functors. The theory also yields spectral sequences relating tendonderived functors to classical ones when a filtration compatible with T exists. In favorable settings, computations can be reduced to standard derived functors using a tendon-admissible resolution.

Applications and limitations: The construction has been explored in representation theory, algebraic geometry, and deformation theory as a way to encode additional structures or symmetries. Critics note that choosing a tendon structure can be non-unique and that computations may become more involved than for standard derived functors. As a result, tendonderived remains a specialized tool used in particular research contexts.

See also: Derived category; Derived functor; Homological algebra; Triangulated category; Spectral sequence.