feat(CategoryTheory): a small colimit of presentable objects is presentable

[joelriou]

In this PR, we show that κ-accessible functors (to the category of types) are stable under limits indexed by a small category K such that Cardinal.mk (Arrow K) < κ.
In particular, κ-presentable objects are stable by colimits indexed by a category K such that HasCardinalLT (Arrow K) κ.

---

• depends on: feat(CategoryTheory): presentable objects #19937
• depends on: feat(CategoryTheory): Arrow A is finite iff A is a finite category #19945
• depends on: feat(SetTheory): the predicate HasCardinalLT #19959
• depends on: feat(CategoryTheory): κ-filtered categories #20005
• depends on: feat(CategoryTheory): the category with two objects zero and one, and an arbitrary type of morphisms from zero to one #20006

[github-actions]

PR summary bcf2d2fef3

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.SetTheory.Cardinal.HasCardinalLT (new file)	772
Mathlib.CategoryTheory.Presentable.ParallelMaps (new file)	800
Mathlib.CategoryTheory.Comma.CardinalArrow (new file)	821
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered (new file)	854
Mathlib.CategoryTheory.Presentable.Basic (new file)	865
Mathlib.CategoryTheory.Presentable.Limits (new file)	871

---

Declarations diff

+ Arrow.discreteEquiv
+ Arrow.equivSigma
+ Arrow.ext
+ Arrow.finite
+ Arrow.finite_iff
+ Arrow.opEquiv
+ Arrow.shrinkEquiv
+ Arrow.shrinkHomsEquiv
+ HasCardinalFilteredColimits
+ HasCardinalLT
+ Hom
+ Hom.comp
+ IsCardinalAccessible
+ IsCardinalFiltered
+ IsCardinalPresentable
+ ParallelMaps
+ arrowEquiv
+ cocone
+ coeq
+ coeqHom
+ coeq_condition
+ hasCardinalLT
+ hasCardinalLT_aleph0_iff
+ hasCardinalLT_arrow_discrete_iff
+ hasCardinalLT_arrow_shrinkHoms_iff
+ hasCardinalLT_arrow_shrink_iff
+ hasCardinalLT_iff_cardinal_mk_lt
+ hasCardinalLT_iff_of_equiv
+ hasCardinalLT_of_hasCardinalLT_arrow
+ hasCardinalLT_option_iff
+ hasCardinal_arrow_op_iff
+ injective
+ instance (C : Type u) [Category.{v} C] [Small.{w'} C] [LocallySmall.{w} C] :
+ instance : Category (ParallelMaps T)
+ instance : Fact Cardinal.aleph0.IsRegular
+ instance : PreservesColimit F (uliftFunctor.{w, w'})
+ instance : PreservesColimitsOfShape J (uliftFunctor.{w, w'})
+ instance : PreservesColimitsOfSize.{v₁, u₁} (uliftFunctor.{w, w'})
+ instance : ReflectsColimit F (uliftFunctor.{w, w'})
+ instance : ReflectsColimitsOfShape J (uliftFunctor.{w, w'})
+ instance : ReflectsColimitsOfSize.{v₁, u₁} (uliftFunctor.{w, w'})
+ instance [HasColimitsOfSize.{w, w} C] : HasCardinalFilteredColimits.{w} C κ
+ isCardinalAccessible_of_isLimit
+ isCardinalAccessible_of_iso
+ isCardinalAccessible_of_le
+ isCardinalFiltered_aleph0_iff
+ isCardinalFiltered_preorder
+ isCardinalPresentable_iff_of_isEquivalence
+ isCardinalPresentable_of_equivalence
+ isCardinalPresentable_of_isColimit
+ isCardinalPresentable_of_isColimit'
+ isCardinalPresentable_of_isEquivalence
+ isCardinalPresentable_of_iso
+ isCardinalPresentable_of_le
+ isCardinalPresentable_shrinkHoms_iff
+ isColimitEquiv
+ isColimitMapCocone
+ isColimitOf'
+ isColimit_cocone_iff
+ isColimit_eq_iff'
+ isFiltered_of_isCardinalDirected
+ locallySmall_of_small_arrow
+ max
+ mkFunctor
+ of_equivalence
+ of_injective
+ of_surjective
+ preservesColimitsOfShape_of_isCardinalAccessible
+ preservesColimitsOfShape_of_isCardinalAccessible_of_essentiallySmall
+ preservesColimitsOfShape_of_isCardinalPresentable
+ preservesColimitsOfShape_of_isCardinalPresentable_of_essentiallySmall
+ quotEquiv
+ quotEquiv_comm
+ small
+ small_of_small_arrow
+ surjective
+ toCoeq
+ toMax
++ of_le

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.

---

Increase in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
1516	2	erw

Current commit bcf2d2fef3
Reference commit d5e4a9182e

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).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• feat(CategoryTheory): presentable objects #19937
• feat(CategoryTheory): Arrow A is finite iff A is a finite category #19945
• feat(SetTheory): the predicate HasCardinalLT #19959
• feat(CategoryTheory): κ-filtered categories #20005
• feat(CategoryTheory): the category with two objects zero and one, and an arbitrary type of morphisms from zero to one #20006
  By Dependent Issues (🤖). Happy coding!