feat(Analysis/Convex/SimplicialComplex): add AbstractSimplicialComplex + constructions

[BoltonBailey]

This adds the concept of abstract simplicial complex (and refactors SimplicialComplex in terms of it).

It also adds constructions that makes it easy to define a simplicial complex for any index family of points which is downward closed and which is affinely independent. I also include a construction of (abstract and geometric) simplicial complexes associated with a SimpleGraph, where vertices become 0-faces and edges become 1-faces, which could be useful later in defining the topological notion of a graph embedding.

Co-authored-by: Claude Opus 4.5

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• depends on: [Merged by Bors] - refactor: reuse IsRelLowerSet in Geometry.SimplicialComplex #35115

[github-actions]

PR summary 738faa5527

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.Convex.SimplicialComplex.Basic	1
Mathlib.AlgebraicTopology.SimplicialComplex.Basic (new file)	1056
Mathlib.Analysis.Convex.SimplicialComplex.AffineIndependentUnion (new file)	1133

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Declarations diff

+ AbstractSimplicialComplex
+ LinearIndependent.affineIndependent
+ PreAbstractSimplicialComplex
+ PreAbstractSimplicialComplex.toAbstractSimplicialComplex
+ PreAbstractSimplicialComplex.toAbstractSimplicialComplex_union_singleton
+ instance : Bot (AbstractSimplicialComplex ι)
+ instance : Bot (PreAbstractSimplicialComplex ι)
+ instance : CompleteLattice (AbstractSimplicialComplex ι)
+ instance : CompleteLattice (PreAbstractSimplicialComplex ι)
+ instance : CompleteSemilatticeInf (AbstractSimplicialComplex ι)
+ instance : CompleteSemilatticeInf (PreAbstractSimplicialComplex ι)
+ instance : CompleteSemilatticeSup (AbstractSimplicialComplex ι)
+ instance : CompleteSemilatticeSup (PreAbstractSimplicialComplex ι)
+ instance : InfSet (AbstractSimplicialComplex ι)
+ instance : InfSet (PreAbstractSimplicialComplex ι)
+ instance : LE (AbstractSimplicialComplex ι)
+ instance : LE (PreAbstractSimplicialComplex ι)
+ instance : LT (AbstractSimplicialComplex ι)
+ instance : LT (PreAbstractSimplicialComplex ι)
+ instance : Max (AbstractSimplicialComplex ι)
+ instance : Max (PreAbstractSimplicialComplex ι)
+ instance : Min (AbstractSimplicialComplex ι)
+ instance : Min (PreAbstractSimplicialComplex ι)
+ instance : PartialOrder (AbstractSimplicialComplex ι)
+ instance : PartialOrder (PreAbstractSimplicialComplex ι)
+ instance : SetLike (AbstractSimplicialComplex ι) (Finset ι)
+ instance : SetLike (PreAbstractSimplicialComplex ι) (Finset ι)
+ instance : SupSet (AbstractSimplicialComplex ι)
+ instance : SupSet (PreAbstractSimplicialComplex ι)
+ instance : Top (AbstractSimplicialComplex ι)
+ instance : Top (PreAbstractSimplicialComplex ι)
+ map
+ ofAffineIndependent
+ onFinsupp
+ toFinset_ne_empty
+ toPreAbstractSimplicialComplex_injective
+ toPreAbstractSimplicialComplex_le_iff
+ toPreAbstractSimplicialComplex_lt_iff
+ top_toPreAbstractSimplicialComplex
++ ofSimpleGraph

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 scripts/declarations_diff.sh contains some details about this script.

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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).

[BoltonBailey]

Here's a thread to discuss the definition change possibility, other than that, I am fine with this either awaiting that or going ahead on its own and letting the refactor happen later
#mathlib4 > Definition of Simplicial Complex

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - refactor: reuse IsRelLowerSet in Geometry.SimplicialComplex #35115
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

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

[BoltonBailey]

Ok, I changed it to use IsRelLowerSet, the predicate not the structure, so that the field is still named faces rather than carrier, which seems appropriate to me, let me know if I should do something different.