feat(Algebra/Category): P.Under ⊤ R has finite limits if P is stable under finite products and equalizers

[chrisflav]

From Pi1.

---

• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): functoriality of coconeLeftOpOfCone and co #36918
• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): reflecting (co)limits and opposites #36909
• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): creation of limits and opposites #37504
• depends on: chore(CategoryTheory/Limits): dualize existence of limits in Over to Under #38012

[github-actions]

PR summary 16cc871c27

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.CategoryTheory.Limits.Constructions.Over.Connected	696	699	+3 (+0.43%)
Mathlib.CategoryTheory.Limits.Constructions.Over.Basic	725	727	+2 (+0.28%)
Mathlib.CategoryTheory.Limits.MorphismProperty	742	744	+2 (+0.27%)

Import changes for all files

Files	Import difference
49 files Mathlib.Algebra.Category.CommAlgCat.FiniteType Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent Mathlib.AlgebraicGeometry.AffineSpace Mathlib.AlgebraicGeometry.AffineTransitionLimit Mathlib.AlgebraicGeometry.AlgClosed.Basic Mathlib.AlgebraicGeometry.Artinian Mathlib.AlgebraicGeometry.Cover.Over Mathlib.AlgebraicGeometry.Geometrically.Integral Mathlib.AlgebraicGeometry.Geometrically.Reduced Mathlib.AlgebraicGeometry.Group.Abelian Mathlib.AlgebraicGeometry.Group.Smooth Mathlib.AlgebraicGeometry.IdealSheaf.IrreducibleComponent Mathlib.AlgebraicGeometry.Modules.Sheaf Mathlib.AlgebraicGeometry.Modules.Tilde Mathlib.AlgebraicGeometry.Morphisms.Etale Mathlib.AlgebraicGeometry.Morphisms.Finite Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified Mathlib.AlgebraicGeometry.Morphisms.Immersion Mathlib.AlgebraicGeometry.Morphisms.Integral Mathlib.AlgebraicGeometry.Morphisms.Proper Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite Mathlib.AlgebraicGeometry.Morphisms.Separated Mathlib.AlgebraicGeometry.Morphisms.SmoothFiber Mathlib.AlgebraicGeometry.Morphisms.Smooth Mathlib.AlgebraicGeometry.Morphisms.WeaklyEtale Mathlib.AlgebraicGeometry.Noetherian Mathlib.AlgebraicGeometry.Normalization Mathlib.AlgebraicGeometry.PointsPi Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper Mathlib.AlgebraicGeometry.QuasiAffine Mathlib.AlgebraicGeometry.RationalMap Mathlib.AlgebraicGeometry.Sites.ElladicCohomology Mathlib.AlgebraicGeometry.Sites.Etale Mathlib.AlgebraicGeometry.Sites.Proetale Mathlib.AlgebraicGeometry.Sites.Small Mathlib.AlgebraicGeometry.SpreadingOut Mathlib.AlgebraicGeometry.ValuativeCriterion Mathlib.AlgebraicGeometry.ZariskisMainTheorem Mathlib.CategoryTheory.Comma.Over.StrictInitial Mathlib.CategoryTheory.MorphismProperty.Comma Mathlib.CategoryTheory.MorphismProperty.OverAdjunction Mathlib.CategoryTheory.Sites.Point.IsMonoidalW Mathlib.CategoryTheory.Sites.Point.Monoidal Mathlib.CategoryTheory.Sites.Point.Over Mathlib.Condensed.Light.InternallyProjective Mathlib.Condensed.Light.Monoidal	1
15 files Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous Mathlib.AlgebraicTopology.ModelCategory.Over Mathlib.CategoryTheory.Limits.Constructions.Over.Basic Mathlib.CategoryTheory.Limits.MorphismProperty Mathlib.CategoryTheory.MorphismProperty.CommaSites Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime Mathlib.CategoryTheory.Sites.Descent.DescentData Mathlib.CategoryTheory.Sites.Descent.IsPrestack Mathlib.CategoryTheory.Sites.Descent.IsStack Mathlib.CategoryTheory.Sites.Descent.Precoverage Mathlib.CategoryTheory.Sites.Monoidal Mathlib.CategoryTheory.Sites.Over Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver Mathlib.CategoryTheory.Sites.SheafHom Mathlib.CategoryTheory.Sites.SubcanonicalOver	2
3 files Mathlib.CategoryTheory.Adhesive.Over Mathlib.CategoryTheory.Limits.Constructions.Over.Connected Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits	3
Mathlib.Algebra.Category.Ring.Under.Property (new file)	2661

---

Declarations diff

+ CreatesFiniteProducts.mk'
+ HasEqualizers
+ HasFiniteProducts
+ StructuredArrow.closedUnderColimitsOfShape_discrete_empty
+ StructuredArrow.isClosedUnderLimitsOfShape
+ Under.closedUnderColimitsOfShape_discrete_empty
+ Under.closedUnderColimitsOfShape_pushout
+ Under.createsEqualizersForget
+ Under.createsFiniteLimitsForget
+ Under.createsFiniteProductsForget
+ Under.hasEqualizers
+ Under.hasFiniteLimits
+ Under.hasFiniteProducts
+ Under.property_limit_of_hasFiniteProducts_of_hasEqualizers
+ createsColimitsOfShapeForgetOfIsConnected
+ createsLimitsOfShape_walkingCospan
+ hasColimitsOfShape_of_isConnected
+ hasColimitsOfShape_opposite_iff
+ hasColimitsOfShape_opposite_opposite_iff
+ hasColimitsOfSize_opposite_iff
+ hasFiniteColimits_opposite_iff
+ hasFiniteLimits_opposite_iff
+ hasFiniteWidePullbacks_opposite_iff
+ hasFiniteWidePushouts_opposite_iff
+ hasLimitsOfShape_opposite_iff
+ hasLimitsOfShape_opposite_opposite_iff
+ hasLimitsOfSize_opposite_iff
+ hasWidePullbacks_opposite_iff
+ hasWidePushouts_opposite_iff
+ instance (priority := 900) [HasPushouts T] [P.IsStableUnderComposition]
+ instance : CreatesFiniteColimits (Under.forget P ⊤ X)
+ instance : PreservesFiniteColimits (Under.forget P ⊤ X)
+ instance [G.Faithful] [G.Full] {Y : A} : HasInitial (StructuredArrow (G.obj Y) G)
+ instance [HasFiniteWidePullbacks C] : HasFiniteWidePushouts Cᵒᵖ := by
+ instance [HasFiniteWidePushouts C] : HasFiniteWidePullbacks Cᵒᵖ := by
+ instance [HasFiniteWidePushouts T] : HasFiniteColimits (P.Under ⊤ X)
+ instance [HasWidePullbacks.{w} C] : HasWidePushouts.{w} Cᵒᵖ := by
+ instance [HasWidePushouts.{w} C] : HasWidePullbacks.{w} Cᵒᵖ := by
+ instance [IsConnected J] {B : D} : CreatesColimitsOfShape J (StructuredArrow.proj B K)
+ instance [IsConnected J] {B : D} : PreservesColimitsOfShape J (StructuredArrow.proj B K) := by
+ instance [P.ContainsIdentities] (Y : P.Under ⊤ X) :
+ instance [P.ContainsIdentities] : HasInitial (P.Under ⊤ X)
+ instance [P.ContainsIdentities] [P.RespectsIso] :
+ instance {B : C} [HasFiniteWidePushouts C] : HasFiniteColimits (Under B) := by
+ instance {B : C} [HasWidePushouts.{w} C] : HasColimitsOfSize.{w, w} (Under B) := by
+ instance {B : D} [IsConnected J] [HasColimitsOfShape J C] :
+ instance {X : T} : HasInitial (Under X) := Under.mkIdInitial.hasInitial
+ inverseImage_op_overObj
+ inverseImage_op_underObj
+ mkIdInitial
+ preservesColimitsOfShape_forget_of_isConnected
+ structuredArrow_iso_iff
+ under_iso_iff

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.

---

Increase in tech debt: (relative, absolute) = (7.00, 0.00)

Current number	Change	Type
6463	7	backward.isDefEq.respectTransparency

Current commit ea5ed7a468
Reference commit 16cc871c27

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).

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - chore(CategoryTheory/Limits): functoriality of coconeLeftOpOfCone and co #36918
• [Merged by Bors] - chore(CategoryTheory/Limits): reflecting (co)limits and opposites #36909
• [Merged by Bors] - chore(CategoryTheory/Limits): creation of limits and opposites #37504
• chore(CategoryTheory/Limits): dualize existence of limits in Over to Under #38012
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.