feat(CategoryTheory/Abelian/TorsionTheory): Introduce torsion theory for abelian categories

[farmanb]

---

[github-actions]

PR summary b6118399e6

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.CategoryTheory.Abelian.TorsionTheory.Basic (new file)	832

---

Declarations diff

+ TorsionTheory
+ eq_rightOrthogonal_leftOrthogonal
+ gc_rightOrthogonal_leftOrthogonal
+ instance (P : ObjectProperty C) : (P.leftOrthogonal).IsClosedUnderExtensions
+ instance (P : ObjectProperty C) : (P.leftOrthogonal).IsClosedUnderQuotients
+ instance (P : ObjectProperty C) {J : Type u'} [Category.{v'} J] :
+ isClosedUnderCoproducts_of_torsionTheory
+ isClosedUnderExtensions_of_torsionTheory
+ isClosedUnderQuotients_of_torsionTheory
+ isLimit_kernelFork_pullbackπ_cokernel_π
+ isTorsionClass_iff
+ kernelForkPullbackπCokernelπ
+ le_leftOrthogonal_rightOrthogonal
+ le_pullback_cokernel_π
+ le_rightOrthogonal_leftOrthogonal
+ le_sSup_of_prop
+ leftOrthogonal_rightOrthogonal_leftOrthogonal_eq_leftOrthogonal
+ leftOrthogonal_rightOrthogonal_leftOrthogonal_eq_rightOrthogonal
+ mem_free_iff
+ rightOrthogonal_cokernel_sSup
+ rightOrthogonal_leftOrthogonal_le
+ sSup_prop
+ shortComplexPullbackπCokernelπ
+ shortExact_shortComplexPullbackπCokernelπ
+ torsionTheoryCogeneratedBy
+ torsionTheoryGeneratedBy

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[farmanb]

WIP

[plp127]

this resembles Concept

[joelriou]

It would be useful to show this:

lemma gc_rightOrthogonal_leftOrthogonal :
    GaloisConnection (OrderDual.toDual (α := ObjectProperty C) ∘ rightOrthogonal)
  (leftOrthogonal ∘ OrderDual.ofDual) :=
  fun _ _ ↦ ⟨fun h _ hX _ _ hY ↦ h _ hY _ hX, fun h _ hX _ _ hY ↦ h _ hY _ hX⟩

[joelriou]

This should hold for colimits of any shape, and this should even be an instance for MorphismProperty.leftOrthogonal (and there is a dual statement for rightOrthogonal).
In general, from the various lemmas, could you extract the statements (instances) which hold for any left or right orthogonal?

[joelriou]

Here, you may probably simplify the proof a little bit by showing that C has coproducts of shape subobjs (and P is stable under such coproducts) by transporting the assumptions through an equivalence of discrete categories (also given by equivShrink, but once these instance are around, equivShrink will no longer be necessary in the rest of the code).

[farmanb]

I took a stab at this. I'm not sure it's actually simpler. I can't see a way to avoid using equivShrink since sSup subobjs is defined to be the image of smallCoproductDesc subobjs, which in turn is defined via equivShrink. The colimit over the subobjs-indexed functor is isomorphic but not definitionally equal to the coproduct inside sSup, so it appears necessary to build the bridge between them explicitly. I'm not sure if you see something that I don't, or if I've misunderstood your intent...

[joelriou]

It is even more simple:

lemma prop_sSup_subobjectOf (P : ObjectProperty C)
    [P.IsClosedUnderQuotients] [∀ J : Type w, P.IsClosedUnderColimitsOfShape (Discrete J)]
    [LocallySmall.{w} C] [WellPowered.{w} C] [HasCoproducts.{w} C]
    (X : C) : P (Subobject.sSup {A : Subobject X | P (A : C)}) :=
  P.prop_of_iso (Subobject.underlyingIso
    (Limits.image.ι (Subobject.smallCoproductDesc _))).symm
      (P.prop_of_epi (factorThruImage _)
        ((ObjectProperty.prop_colimit _ _ (fun ⟨j⟩ ↦ by
          dsimp
          obtain ⟨S, hS, hj⟩ := j.2
          simpa [← hj] using hS))))