feat(AlgebraicTopology): homology in degree 0

[joelriou]

In this PR, we compute the singular homology of a topological space in degree 0 in terms of its path connected components. This follows from a similar result for the homology of simplicial sets.

This is a rough draft.

---

• depends on: feat(AlgebraicTopology): simplicial homology #37656
• depends on: feat(CategoryTheory/Limits): sigmaConst preserves colimits #37645
• depends on: [Merged by Bors] - feat(AlgebraicTopology/SimplicialSet): connected components of simplicial sets #37607

[github-actions]

PR summary 6b3e5a3312

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.AlgebraicTopology.SingularHomology.HomotopyInvariance	-1710
7 files Mathlib.Algebra.Category.ModuleCat.Sheaf.Free Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent Mathlib.AlgebraicGeometry.Modules.Tilde Mathlib.AlgebraicTopology.SingularHomology.Basic Mathlib.AlgebraicTopology.SingularHomology.HomotopyInvarianceTopCat	1
Mathlib.AlgebraicTopology.SimplicialSet.PiZero	3
Mathlib.CategoryTheory.Limits.Preserves.SigmaConst (new file)	466
Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic (new file)	1086
Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomologyZero (new file)	1091
Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomotopyInvariance	1137
Mathlib.AlgebraicTopology.SingularHomology.HomologyZero (new file)	1715

---

Declarations diff

+ Homeomorph.continousMap
+ Path
+ TopCat.stdSimplexHomeomorphI_symm_one
+ TopCat.stdSimplexHomeomorphI_symm_zero
+ TopCat.stdSimplexHomeomorphI_vertex_one
+ TopCat.stdSimplexHomeomorphI_vertex_zero
+ chainComplex
+ chainComplexFunctor
+ chainComplexFunctorAdjunction
+ chainComplexXCofan
+ chainComplex_hom_ext
+ coforkπ₀
+ coforkπ₀Functor
+ congr_homologyMap
+ congr_sSetHomologyMap
+ freeFunctor_map
+ freeFunctor_obj
+ homology
+ homologyData₀
+ homologyFunctor
+ homologyMap
+ homologyMap_comp
+ homologyMap_id
+ homology₀Iso
+ homology₀ε
+ instance :
+ instance : (chainComplexFunctor C).Additive := by
+ instance : Unique (stdSimplex ℝ (Fin (⦋0⦌.len + 1)))
+ instance [HasCoproducts.{w} C] (R : C) :
+ instance [HasCoproducts.{w} C] {α β : Type w} (f : α → β) :
+ instance [PathConnectedSpace X] : (toSSet.obj X).IsConnected := by
+ instance [PathConnectedSpace X] : IsIso (X.singularHomology₀ε R)
+ instance [PathConnectedSpace X] : Nonempty (ZerothHomotopy X)
+ instance [PathConnectedSpace X] : Subsingleton (ZerothHomotopy X)
+ instance [X.IsConnected] : IsIso (X.homology₀ε R) := by
+ instance [X.IsConnected] : Nonempty (π₀ X) := ⟨π₀.mk (Classical.arbitrary _)⟩
+ isColimitChainComplexXCofan
+ isColimitCoforkπ₀
+ isColimitCoforkπ₀Functor
+ isColimitCokernelCoforkChainComplexDOneZero
+ isColimitSigmaConstCokernelCofork
+ isConnected_iff
+ isConnected_iff_nonempty_unique
+ lift
+ lift_mk
+ mk
+ mk_surjective
+ pathEquiv
+ rec
+ sSetChainComplexMap
+ sigmaConstCokernelCofork
+ singularHomology₀Iso
+ singularHomology₀Iso_sigma_desc_id
+ singularHomology₀ε
+ sound
+ toSSetObjEdgeEquiv
+ toSSetObjEdgeEquiv'
+ toSSetObjEquiv_naturality_apply
+ toSSetObjEquiv_symm_naturality
+ toSSetObjEquiv_δ_apply
+ toSSetObj₁Equiv
+ toSSetObj₁Equiv_apply_one
+ toSSetObj₁Equiv_apply_zero
+ toπ₀NatTrans
+ zerothHomotopyEquiv
+ δ_one_toSSetObj₁Equiv.symm
+ δ_zero_toSSetObj₁Equiv.symm
+ ιChainComplex
+ ιChainComplex_d
+ ι_chainComplexMap_f
+ ι_sigmaConstCokernelCofork_π
+ ι_sigmaConstCokernelCofork_π_eq_zero
+ π₀.comp_fromChainComplexXZero
+ π₀.d_fromChainComplexXZero
+ π₀.fromChainComplexXZero
+ π₀.nonempty_iff
++ chainComplexMap
++-- congr_homologyMap_singularChainComplexFunctor
++-- singularChainComplexFunctorObjMap
- SSet.singularChainComplexFunctor
- SSet.singularChainComplexFunctorAdjunction
- instance : (SSet.singularChainComplexFunctor C).Additive := by

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) = (11.00, 0.00)

Current number	Change	Type
6779	11	backward.isDefEq.respectTransparency

Current commit e97d90992f
Reference commit 6b3e5a3312

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).

---

note: file Mathlib/AlgebraicTopology/SingularHomology/HomotopyInvariance.lean` was renamed to `Mathlib/AlgebraicTopology/SimplicialSet/Homology/HomotopyInvariance.lean without a module deprecation
Please create a follow-up pull request adding one. Thanks!

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(AlgebraicTopology): simplicial homology #37656
• feat(CategoryTheory/Limits): sigmaConst preserves colimits #37645
• [Merged by Bors] - feat(AlgebraicTopology/SimplicialSet): connected components of simplicial sets #37607
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

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