Foundation Singular Subdivision T Op Iter Chain Homotopy Zero

The subdivision homotopy

In algebraic topology, a chain complex is a sequence of abelian groups connected by boundary maps, and a chain map between complexes is a structure-preserving translation. Subdivision is a way to replace a space by a finer one, cutting each simplex into smaller simplices, and it appears throughout the subject, for example in proofs of invariance theorems. The framework's machine-checked library of formal theorems records a precise statement about the interaction of subdivision with boundaries.

The declaration tOp_chain_homotopy_zero is a theorem in the framework's library. It states that for any topological space X, the map tOp X 0 composed with the boundary map bnd X 0 equals the identity on the zero-dimensional chain group minus the subdivision operator sdOp X 0. In plain terms, the first subdivision step, followed by taking the boundary, differs from doing nothing by exactly the subdivision operator itself. This is a chain homotopy statement: it says the two maps are homotopic as chain maps, which is a standard way to show they induce the same map on homology.

The theorem is proved by a short calculation: it rewrites tOp X 0 to zero, sdOp X 0 to the identity, and then uses the fact that zero composed with anything is zero, so the right side becomes identity minus identity, which is zero. The declaration is tagged as a theorem in the library, meaning it is a machine-checked result with no unproved assumptions beyond the standard axioms of the ambient type theory.

What the declaration does not claim is broader. It does not assert that subdivision is a homotopy equivalence in general, nor that the chain homotopy exists for higher dimensions. The theorem covers only the case n = 0, the zero-dimensional chains. It also does not claim anything about the geometric content of subdivision, such as how barycentric coordinates behave, only the algebraic relation between the operators. The statement is one small step in a larger development of singular subdivision, not a standalone result about the topology of spaces.

/-- In degree `0`: `∂ ∘ T = id − S` (both sides vanish). -/
theorem tOp_chain_homotopy_zero (X : TopCat.{0}) :
    tOp X 0 ≫ bnd X 0 = 𝟙 (Cgrp X 0) - sdOp X 0 := by
  rw [tOp_zero, sdOp_zero, Limits.zero_comp, sub_self]

/-- In degree `0`: `∂ ∘ T = id − S` (both sides vanish). -/
theorem tOp_chain_homotopy_zero (X : TopCat.{0}) :
    tOp X 0 ≫ bnd X 0 = 𝟙 (Cgrp X 0) - sdOp X 0 := by
  rw [tOp_zero, sdOp_zero, Limits.zero_comp, sub_self]

/-- In degree `0`: `∂ ∘ T = id − S` (both sides vanish). -/
theorem tOp_chain_homotopy_zero (X : TopCat.{0}) :
    tOp X 0 ≫ bnd X 0 = 𝟙 (Cgrp X 0) - sdOp X 0 := by
  rw [tOp_zero, sdOp_zero, Limits.zero_comp, sub_self]

What this page does not claim

The theorem does not assert that subdivision is a homotopy equivalence in general. The theorem does not claim the chain homotopy exists for dimensions higher than zero. The theorem does not describe the geometric behavior of barycentric coordinates.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/SingularSubdivision.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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