refactor(AlgebraicTopology/SimplicialSet): use the Subpresheaf API

[joelriou]

The type of subcomplexes SSet.Subcomplex of a simplicial set is defined as an abbreviation for Subpresheaf, and boundaries and horns are redefined as subcomplexes of the standard simplex.

• depends on: feat(Order): Fin.succAboveOrderIso #21303
• depends on: [Merged by Bors] - feat(CategoryTheory/Subpresheaf): preimage/image/range of subpresheaves #21047
• depends on: [Merged by Bors] - feat(CategoryTheory): Subpresheaf is a complete lattice #20840
• depends on: [Merged by Bors] - feat(CategoryTheory/Subpresheaf): equalizer #21096

---

[github-actions]

PR summary b57907dbe6

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.AlgebraicTopology.SimplicialSet.Horn	842	850	+8 (+0.95%)
Mathlib.AlgebraicTopology.SimplicialSet.KanComplex	843	851	+8 (+0.95%)
Mathlib.AlgebraicTopology.SimplicialSet.Path	843	851	+8 (+0.95%)
Mathlib.AlgebraicTopology.Quasicategory.Basic	844	852	+8 (+0.95%)
Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal	849	857	+8 (+0.94%)
Mathlib.AlgebraicTopology.Quasicategory.StrictSegal	852	860	+8 (+0.94%)
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex	841	848	+7 (+0.83%)
Mathlib.AlgebraicTopology.SimplicialSet.Boundary	842	849	+7 (+0.83%)
Mathlib.AlgebraicTopology.ExtraDegeneracy	1071	1078	+7 (+0.65%)

Import changes for all files

Files	Import difference
4 files Mathlib.RepresentationTheory.GroupCohomology.Basic Mathlib.RepresentationTheory.GroupCohomology.Hilbert90 Mathlib.RepresentationTheory.GroupCohomology.LowDegree Mathlib.RepresentationTheory.GroupCohomology.Resolution	6
3 files Mathlib.AlgebraicTopology.ExtraDegeneracy Mathlib.AlgebraicTopology.SimplicialSet.Boundary Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex	7
9 files Mathlib.AlgebraicTopology.Quasicategory.Basic Mathlib.AlgebraicTopology.Quasicategory.Nerve Mathlib.AlgebraicTopology.Quasicategory.StrictSegal Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat Mathlib.AlgebraicTopology.SimplicialSet.Horn Mathlib.AlgebraicTopology.SimplicialSet.KanComplex Mathlib.AlgebraicTopology.SimplicialSet.Path Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal	8
Mathlib.Order.Fin.SuccAboveOrderIso (new file)	457
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex (new file)	838

---

Declarations diff

+ Fin.succAboveOrderIso
+ Subcomplex
+ Subcomplex.liftPath
+ Subcomplex.map_ι_liftPath
+ Subcomplex.toSSet
+ Subcomplex.ι
+ asOrderHom
+ comp_const
+ const_comp
+ const_val_apply
+ ext
+ faceRepresentableBy
+ face_eq_ofSimplex
+ face_inter_face
+ face_le_subcomplexHorn
+ instance (A : X.Subcomplex) :
+ instance (n i : ℕ) : DFunLike (Δ[n] _[i]) (Fin (i + 1)) (fun _ ↦ Fin (n + 1))
+ instance : CoeOut X.Subcomplex SSet.{u}
+ isLocallySurjective_iff_range_sheafify_eq_top
+ isLocallySurjective_iff_range_sheafify_eq_top'
+ isoOfRepresentableBy
+ mem_face_iff
+ mem_ofSimplex_obj
+ objEquiv_symm_apply
+ objEquiv_symm_comp
+ objEquiv_toOrderHom_apply
+ objMk_apply
+ obj₀Equiv
+ ofSimplex
+ ofSimplex_le_iff
+ ofSimplex_yonedaEquiv_δ
+ range_eq_ofSimplex
+ range_δ
+ stdSimplex.spineId_arrow_apply_one
+ stdSimplex.spineId_arrow_apply_zero
+ stdSimplex.spineId_vertex
+ subcomplexBoundary
+ subcomplexBoundary_eq_iSup
+ subcomplexHorn
+ subcomplexHorn.spineId
+ subcomplexHorn.spineId_map_hornInclusion
+ subcomplexHorn_eq_iSup
+ subcomplexHorn_obj_zero
+ succ_succAbove_predAbove
+ yonedaEquiv_coe
+ yonedaEquiv_comp
+ yonedaEquiv_map
++- face
+-+ const

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.

---

No changes to technical debt.

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(Order): Fin.succAboveOrderIso #21303
• [Merged by Bors] - feat(CategoryTheory/Subpresheaf): preimage/image/range of subpresheaves #21047
• [Merged by Bors] - feat(CategoryTheory): Subpresheaf is a complete lattice #20840
• [Merged by Bors] - feat(CategoryTheory/Subpresheaf): equalizer #21096
  By Dependent Issues (🤖). Happy coding!

[gio256]

Would it be useful to add these to a new namespace, SSet.subcomplex.horn and SSet.subcomplex.boundary? Potentially the lemmas could also then become, e.g., SSet.subcomplex.horn.eq_iSup and SSet.subcomplex.boundary.eq_iSup.

[gio256]

This went from an ugly proof to an elegant one. Thanks!