homology f g w, where
w : f ≫ g = 0, can be identified with either a
cokernel or a kernel. The isomorphism with a cokernel is
was obtained elsewhere. In the case of an abelian category, this file shows the isomorphism
with a kernel as well.
We use these isomorphisms to obtain the analogous api for
homology.ιis the map from
homology f g winto the cokernel of
homology.π'is the map from
homology f g w.
homology.desc'constructs a morphism from
homology f g w, when it is viewed as a cokernel.
homology.liftconstructs a morphism to
homology f g w, when it is viewed as a kernel.
- Various small lemmas are proved as well, mimicking the API for (co)kernels. With these definitions and lemmas, the isomorphisms between homology and a (co)kernel need not be used directly.
The homology associated to
g is isomorphic to a kernel.
The canonical map from the kernel of
g to the homology of
The canonical map from the homology of
g to the cokernel of
Obtain a morphism from the homology, given a morphism from the kernel.
Obtain a moprhism to the homology, given a morphism to the kernel.
F is an exact additive functor,
F(Hᵢ(X)) ≅ Hᵢ(F(X)) for
X a complex.
F is an exact additive functor, then
F commutes with
Hᵢ (up to natural isomorphism).