feat(CategoryTheory/Sites): local sites

[peabrainiac]

Defines local sites and shows that sheaves of types on them form a local topos, in that the global sections functor Sheaf.Γ has a right adjoint Sheaf.codisc that is fully faithful.

---

• depends on: [Merged by Bors] - feat(CategoryTheory/Sites): global sections functor on sheaves #22816

[github-actions]

PR summary b4f07ded54

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Products	423	426	+3 (+0.71%)

Import changes for all files

Files	Import difference
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Products	3
Mathlib.CategoryTheory.Sites.GlobalSections (new file)	962
Mathlib.CategoryTheory.Sites.LocalSite (new file)	967

---

Declarations diff

+ LocalSite
+ LocalSite.from_terminal_mem_of_mem
+ Presheaf.codisc
+ Presheaf.codisc_isSheaf
+ Sheaf.codisc
+ Sheaf.codiscΓNatIsoId
+ Sheaf.fullyFaithfulCodisc
+ Sheaf.Γ
+ Sheaf.ΓCodiscAdj
+ Sheaf.ΓNatIsoCoyonedaObj
+ Sheaf.ΓNatIsoSectionsFunctor
+ Sheaf.ΓNatIsoSheafSections
+ constantSheafΓAdj
+ fullyFaithfulConstantSheaf
+ instance [HasWeakSheafify J A] [HasLimitsOfShape Cᵒᵖ A] : (constantSheaf J A).IsLeftAdjoint
+ instance [HasWeakSheafify J A] [HasLimitsOfShape Cᵒᵖ A] : (Γ J A).IsRightAdjoint
+ instance [LocalSite J] : (codisc.{u,v,max u v w} J).Faithful
+ instance [LocalSite J] : (codisc.{u,v,max u v w} J).Full
+ instance [LocalSite J] : (codisc.{u,v,max u v w} J).IsRightAdjoint
+ instance [LocalSite J] : (Γ J (Type max u v w)).IsLeftAdjoint
+ instance {C : Type u} [Category.{v} C] [HasTerminal C] : LocalSite (trivial C)
+ limNatIsoSectionsFunctor
+ sectionsFunctorNatIsoCoyoneda
+ sheafSectionsNatIsoEvaluation
+ terminalObjIso
++ instance [HasWeakSheafify J (Type max u v w)] [LocalSite J] :

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[joelriou]

As this is a Prop, we usually like to use the prefix Is. This should probably go into the GrothendieckTopology namespace, so that we may write J.IsLocal, or J.IsLocalSite.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(CategoryTheory/Sites): global sections functor on sheaves #22816
  By Dependent Issues (🤖). Happy coding!

[joelriou]

Could you add a reference where this notion of local site is used?

[peabrainiac]

Migrated to #41083.