CokerT

CokerT is a term used in category theory and homological algebra to denote the cokernel of a morphism after applying a functor T. Given a morphism f: A → B in a category C and a functor T: C → D, cokerT(f) refers to the cokernel of the induced morphism T(f): T(A) → T(B) in the target category D. The construction depends on T and on the ambient category D; in practice, D is taken to be an abelian category so that cokernels exist for all morphisms. When T preserves cokernels, there is a natural isomorphism T(coker(f)) ≅ coker(T(f)).

Notation and usage vary; some authors use coker_T(f), coker(Tf), or coker^T(f). Because cokerT is not standardized, the precise meaning should be inferred from context, especially the definition of T and the category D.

Relationship to exactness: If f is part of a short exact sequence and T is exact, then applying T preserves exactness at the cokernel, and coker(T(f)) reflects properties of coker(f). This concept is relevant in the study of functorial behavior of exact sequences and in the construction of derived functors.

Example: Let f: A → B in Ab and T is the identity functor; cokerT(f) = coker(f).

See also: cokernel, functor, abelian category, exact functor, homological algebra.