Exact sequences #
In a category with zero morphisms, images, and equalizers we say that
f : A ⟶ B and
g : B ⟶ C
are exact if
f ≫ g = 0 and the natural map
image f ⟶ kernel g is an epimorphism.
In any preadditive category this is equivalent to the homology at
However in general it is weaker than other reasonable definitions of exactness, particularly that
- the inclusion map
image.ι fis a kernel of
image f ⟶ kernel gis an isomorphism or
image_subobject f = kernel_subobject f. However when the category is abelian, these all become equivalent; these results are found in
Main results #
- Suppose that cokernels exist and that
gare exact. If
sis any kernel fork over
tis any cokernel cofork over
fork.ι s ≫ cofork.π t = 0.
- Precomposing the first morphism with an epimorphism retains exactness. Postcomposing the second morphism with a monomorphism retains exactness.
gare exact and
iis an isomorphism, then
f ≫ i.homand
i.inv ≫ gare also exact.
Future work #
- Short exact sequences, split exact sequences, the splitting lemma (maybe only for abelian categories?)
- Two adjacent maps in a chain complex are exact iff the homology vanishes
f : A ⟶ B,
g : B ⟶ C are called exact if
w : f ≫ g = 0 and the natural map
image_to_kernel f g w : image_subobject f ⟶ kernel_subobject g is an epimorphism.
In any preadditive category, this is equivalent to
w : f ≫ g = 0 and
homology f g w ≅ 0.
In an abelian category, this is equivalent to
image_to_kernel f g w being an isomorphism,
and hence equivalent to the usual definition,
image_subobject f = kernel_subobject g.
In any preadditive category,
f g are exact iff they compose to zero and the homology vanishes.