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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,

Relationship to exactness: If f is part of a short exact sequence and T is exact, then

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.

the
precise
meaning
should
be
inferred
from
context,
especially
the
definition
of
T
and
the
category
D.
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.