feat(CategoryTheory/Limits): commutation of limits and colimits

[joelriou]

This PR adds some API about the commutation of limits and colimits. For example, if C has colimits of shape K' and limits of shape K, we show that the limit functor lim : (K ⥤ C) ⥤ C commutes with colimits of shape K' iff the colimit functor colim : (K' ⥤ C) ⥤ C commutes with limits of shape K.
We use this in order to prove that if C has exact colimits of shape K' then the full subcategory of J ⥤ C consisting of functors which commutes with colimits of shape K' is closed under finite limits and finite colimits.

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I do not think it would make sense to split the file Mathlib/CategoryTheory/ObjectProperty/FunctorCategory/PreservesLimits.lean to reduce imports.

[github-actions]

PR summary 61a2abceca

Import changes exceeding 2%

%	File
+16.04%	Mathlib.CategoryTheory.ObjectProperty.FunctorCategory.PreservesLimits

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.CategoryTheory.ObjectProperty.FunctorCategory.PreservesLimits	1016	1179	+163 (+16.04%)

Import changes for all files

Files	Import difference
Mathlib.CategoryTheory.Limits.Types.PreservesLimit Mathlib.CategoryTheory.ObjectProperty.FunctorCategory.PreservesLimits	163

---

Declarations diff (regex)

+ colim.coconeFlip
+ colim.isColimitCoconeFlip
+ colimitLimToLimitColim
+ colimitLimToLimitColim_eq_colimit_desc
+ colimitLimToLimitColim_eq_limit_lift
+ instance (F : K ⥤ J) [HasColimitsOfShape K' C] :
+ instance [HasColimitsOfShape K' C] :
+ instance [HasColimitsOfShape K' C] [HasExactColimitsOfShape K' C] [HasFiniteLimits C] :
+ instance [HasColimitsOfShape K' C] [HasLimitsOfShape K C]
+ instance [HasFiniteColimits C] :
+ instance {ι : Type*} (P : ι → ObjectProperty C) [∀ i, (P i).IsClosedUnderColimitsOfShape J] :
+ instance {ι : Type*} (P : ι → ObjectProperty C) [∀ i, (P i).IsClosedUnderLimitsOfShape J] :
+ isIso_colimitLimToLimitColim_iff_preservesColimit
+ isIso_colimitLimToLimitColim_iff_preservesLimit
+ lim.cone
+ lim.isLimitCone
+ preservesColimit_flip_lim_iff_preservesLimit_colim
+ preservesColimit_lim_iff_preservesLimit_colim
+ preservesColimitsOfShape_lim_iff_preservesLimitsOfShape_colim
+ ι_colimitToLimit_π

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 61a2abc).

• +25 new declarations
• −0 removed declarations

+CategoryTheory.Limits.colim.coconeFlip
+CategoryTheory.Limits.colim.coconeFlip_pt
+CategoryTheory.Limits.colim.coconeFlip_ι_app_app
+CategoryTheory.Limits.colim.isColimitCoconeFlip
+CategoryTheory.Limits.colimitLimToLimitColim
+CategoryTheory.Limits.colimitLimToLimitColim_eq_colimit_desc
+CategoryTheory.Limits.colimitLimToLimitColim_eq_limit_lift
+CategoryTheory.Limits.isIso_colimitLimToLimitColim_iff_preservesColimit
+CategoryTheory.Limits.isIso_colimitLimToLimitColim_iff_preservesLimit
+CategoryTheory.Limits.lim.cone
+CategoryTheory.Limits.lim.cone_pt
+CategoryTheory.Limits.lim.cone_π_app_app
+CategoryTheory.Limits.lim.isLimitCone
+CategoryTheory.Limits.preservesColimit_flip_lim_iff_preservesLimit_colim
+CategoryTheory.Limits.preservesColimit_lim_iff_preservesLimit_colim
+CategoryTheory.Limits.preservesColimitsOfShape_lim_iff_preservesLimitsOfShape_colim
+CategoryTheory.Limits.ι_colimitToLimit_π
+CategoryTheory.Limits.ι_colimitToLimit_π_assoc
+CategoryTheory.ObjectProperty.instIsClosedUnderColimitsOfShapeFunctorPreservesColimitOfHasColimitsOfShape
+CategoryTheory.ObjectProperty.instIsClosedUnderColimitsOfShapeFunctorPreservesColimitsOfShapeOfHasColimitsOfShape
+CategoryTheory.ObjectProperty.instIsClosedUnderColimitsOfShapeIInf
+CategoryTheory.ObjectProperty.instIsClosedUnderFiniteColimitsFunctorPreservesColimitsOfShapeOfHasFiniteColimits
+CategoryTheory.ObjectProperty.instIsClosedUnderFiniteLimitsFunctorPreservesColimitsOfShapeOfHasExactColimitsOfShapeOfHasFiniteLimits
+CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeFunctorPreservesColimitsOfShapeOfHasLimitsOfShapeOfPreservesLimitsOfShapeColim
+CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeIInf

---

Decrease in strong tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (strong)
backward.defeqAttrib.useBackward	4142	-1

No changes to weak technical debt.

Current commit 61a2abceca
Reference commit 2ed733ad3e

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.py 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).