refactor(CategoryTheory/SmallObject): generalization of the definitions

[joelriou]

The ongoing definition of the iteration of a natural transformation ε : 𝟭 C ⟶ F (with F : C ⥤ C) is generalized to "successor structures" (which shall become a mathlibism), i.e. in a category D, this consists of a zeroth object X₀, a successor application succ : D → D and a map toSucc X : X → succ X (which does not have to be natural: it is not always so in some applications). For such a Φ : SuccStruct D, if J is a well-ordered type, we define the J-th iteration of Φ. (In the case J := ℕ, this is the colimit of succ (succ (succ (succ ... X₀))).)

The iteration of a functor is a particular case of this constructor with D := C ⥤ C.

As toSucc does not have to be natural in X, the caveat is that the proofs make extensive use of equalities of objects in C, while my previous construction used comparison isomorphisms. Nevertheless, the proofs look much more clean now. One of the reasons is that in the inductive construction (file Iteration.Nonempty), in the terms of data, we only need to provide a functor, and then all the fields are in Prop.

This PR supersedes #19264, and overlaps a little bit over the content of #20142.

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[github-actions]

PR summary 6e71693232

Import changes exceeding 2%

%	File
+3.04%	Mathlib.CategoryTheory.SmallObject.Iteration.Basic
+3.04%	Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc
+3.04%	Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone
+3.26%	Mathlib.CategoryTheory.SmallObject.Iteration.Nonempty

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.CategoryTheory.SmallObject.Iteration.UniqueHom	428	0	-428 (-100.00%)
Mathlib.CategoryTheory.SmallObject.Iteration.Nonempty	429	443	+14 (+3.26%)
Mathlib.CategoryTheory.SmallObject.Iteration.Basic	427	440	+13 (+3.04%)
Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc	428	441	+13 (+3.04%)
Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone	428	441	+13 (+3.04%)

Import changes for all files

Files	Import difference
Mathlib.CategoryTheory.SmallObject.Iteration.UniqueHom	-428
3 files Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone Mathlib.CategoryTheory.SmallObject.Iteration.Basic	13
Mathlib.CategoryTheory.SmallObject.Iteration.Nonempty	14
Mathlib.CategoryTheory.SmallObject.Iteration.Iteration (new file)	444

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Declarations diff

+ HasIterationOfShape
+ MapEq
+ SuccStruct
+ congr_colimit_ι
+ congr_map
+ congr_obj
+ congr_toSucc
+ contains
+ extendToSucc_map
+ extendToSucc_obj_eq
+ extendToSucc_obj_succ_eq
+ functor
+ functor_eq
+ functor_map
+ functor_map_to_top
+ functor_obj
+ functor_obj_eq_colimit
+ hasColimitOfShape_of_isSuccLimit
+ inductiveSystem
+ isColimit
+ isColimitIterationCocone
+ isoBot
+ isoSucc
+ iter
+ iteration
+ iterationCocone
+ iterationCocone_pt
+ iterationFunctor
+ mapEq_of_eq
+ mapEq_rfl
+ mapEq_trans
+ mapObj
+ mapObj_refl
+ mapObj_trans
+ map_succ'
+ nonempty
+ obj_eq
+ obj_succ_eq
+ ofCocone_obj_eq
+ ofCocone_obj_eq_pt
+ ofNatTrans
+ restrictionLT_functor
+ restrictionLT_functor_of_lt
+ subsingleton
- Hom
- comp
- congr_app
- eval
- ext'
- extentToSucc_map
- id
- instance : Category (Iteration ε j)
- instance : Nonempty (iter₁ ⟶ iter₂) := by
- instance : Unique (iter₁ ⟶ iter₂)
- instance {J} {j : J} [PartialOrder J] [OrderBot J] [WellFoundedLT J] [SuccOrder J]
- iso
- iso_refl
- iso_trans
- mapSucc
- mapSucc'
- mapSucc_eq
- mkOfLimitNatTrans
- mkOfLimitNatTransApp
- mkOfLimitNatTransApp_eq_of_lt
- mkOfLimitNatTransApp_naturality_top
- mkOfSuccNatTrans
- mkOfSuccNatTransApp
- mkOfSuccNatTransApp_eq_of_le
- mkOfSuccNatTransApp_succ_eq
- natTrans_comp
- natTrans_id
- natTrans_naturality
- truncFunctor
- truncFunctor_map_natTrans_app
-+- mkOfSucc
--+ mkOfBot

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.

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Decrease in tech debt: (relative, absolute) = (4.00, 0.00)

Current number	Change	Type
1510	-4	erw

Current commit 6e71693232
Reference commit 5f24fc48e5

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).