Refinements #
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism X ⟶ Y in an abelian category C is a
monomorphism if and only if for all A : C, the induced map
(A ⟶ X) → (A ⟶ Y) of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends X to the
presheaf of maps A ⟶ X for all A : C preserves and reflects
monomorphisms.
However, if p : X ⟶ Y is an epimorphism in C and A : C,
(A ⟶ X) → (A ⟶ Y) may fail to be surjective (unless p is a split
epimorphism).
In this file, the basic result is epi_iff_surjective_up_to_refinements
which states that f : X ⟶ Y is a morphism in an abelian category,
then it is an epimorphism if and only if for all y : A ⟶ Y,
there exists an epimorphism π : A' ⟶ A and x : A' ⟶ X such
that π ≫ y = x ≫ f. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to Y to a morphism to X.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism π ≫ y is a refinement of y. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: indeed,
show that it corresponds to the "refinements" topology on an
abelian category C that is defined by saying that
a sieve is covering if it contains an epimorphism).
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see ShortComplex.exact_iff_exact_up_to_refinements).
As it is outlined in the documentation of the file
CategoryTheory.Abelian.Pseudoelements, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor ι
from an abelian category C to the category of abelian groups. If we
define a pseudo-element of X : C to be an element in ι.obj X, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
CategoryTheory.Abelian.Pseudoelements. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism X ⟶ Y in C is not determined by
its action on pseudo-elements (see also Counterexamples/Pseudoelement).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms A ⟶ X in C which we may consider as being
"sort of elements of X". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism A' ⟶ A.
References #
- George Bergman, A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups (1974) http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf