feat(AlgebraicTopology/SingularHomology): 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(Topology): more API for ZerothHomotopy #38201
• depends on: [Merged by Bors] - feat(AlgebraicTopology/SimplicialSet): homology in degree 0 #38134
• depends on: [Merged by Bors] - feat(AlgebraicTopology): simplicial homology #37656
• depends on: [Merged by Bors] - 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 2ea7849dc0

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.AlgebraicTopology.SingularHomology.HomologyZero (new file)	1723

---

Declarations diff

+ Homeomorph.continousMap
+ Path
+ TopCat.stdSimplexHomeomorphI_symm_one
+ TopCat.stdSimplexHomeomorphI_symm_zero
+ TopCat.stdSimplexHomeomorphI_vertex_one
+ TopCat.stdSimplexHomeomorphI_vertex_zero
+ instance : Unique (stdSimplex ℝ (Fin (⦋0⦌.len + 1)))
+ 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)
+ lift
+ lift_mk
+ mk
+ mk_surjective
+ pathEquiv
+ rec
+ 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
+ zerothHomotopyEquiv
+ δ_one_toSSetObj₁Equiv.symm
+ δ_zero_toSSetObj₁Equiv.symm

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) = (4.00, 0.00)

Current number	Change	Type
6539	4	backward.isDefEq.respectTransparency

Current commit d57a7341b8
Reference commit 2ea7849dc0

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-merge-conflicts]

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

[mathlib-merge-conflicts]

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

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(Topology): more API for ZerothHomotopy #38201
• [Merged by Bors] - feat(AlgebraicTopology/SimplicialSet): homology in degree 0 #38134
• [Merged by Bors] - feat(AlgebraicTopology): simplicial homology #37656
• [Merged by Bors] - 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.