Foundation Singular Sphere Is Iso Homology Map Aug To

The augmentation map

In algebraic topology, the zeroth homology group of a space records its connected components. For a path-connected space, there is exactly one component, so the group is isomorphic to the integers. The theorem isIso_homologyMap_augTo states that a particular map, called the augmentation map, induces this isomorphism on the zeroth homology group. The augmentation map sends each 0-simplex, which is just a point, to 1 if the point lies in a chosen clopen set, and to 0 otherwise. A clopen set is both open and closed, so its boundary is empty; this is what makes the map compatible with the homology boundary operator.

The theorem is proved in the machine-checked library of formal theorems. It requires the space to be path-connected. The result is that the map from the zeroth homology group of the space to the integers, induced by the augmentation map, is an isomorphism. This means the two groups are structurally identical: every integer arises from some homology class, and different classes give different integers. The proof uses the fact that in a path-connected space, any two points can be joined by a path, so the constant simplex at one point is homologous to the constant simplex at another.

In Recognition Science, this theorem is part of the toolkit for building the singular sphere. The framework models spaces as simplicial sets, where the zeroth homology group plays the role of a discrete ledger of connected components. The theorem ensures that for a path-connected space, this ledger is exactly the integers, with the augmentation map providing the counting function. This is a structural fact about homology, not a claim about physics. It does not say anything about the cost function, the golden ratio, or the forcing chain. It is a purely topological statement about the relationship between a space and its zeroth homology group.

The theorem also does not claim that the augmentation map itself is an isomorphism on all homology groups. It only concerns the zeroth group. Higher homology groups can be nontrivial and are not addressed by this result. The theorem is conditional on the space being path-connected; for a space with multiple components, the zeroth homology group would be a direct sum of integers, one for each component, and the augmentation map would not be an isomorphism.

/-- **`H₀` of a path-connected space.** The augmentation induces an
isomorphism `H₀(X) ≅ ℤ` on homology. -/
theorem isIso_homologyMap_augTo (X : TopCat.{0}) [PathConnectedSpace X] :
    IsIso (HomologicalComplex.homologyMap
      (augTo X Set.univ isClopen_univ) 0) := by
  obtain ⟨x₀⟩ : Nonempty ↥X := PathConnectedSpace.nonempty
  apply isIso_homologyMap_chain_zero
  · intro y
    refine ⟨gen X 0 (constSimplex X x₀) (show ℤ from y), 0, ?_⟩
    rw [Zsingle_d_one_zero, zeroApp, add_zero, augTo_f_zero_apply,
      ← ModuleCat.comp_apply, gen_augFun, if_pos (Set.mem_univ _)]
    exact ModuleCat.id_apply _ _
  · intro z hz
    obtain ⟨w, hw⟩ := hz
    have hz0 : augFun X Set.univ z = 0 := by
      rw [augTo_f_zero_apply, Zsingle_d_one_zero, zeroApp] at hw
      exact hw
    obtain ⟨v, hv⟩ := exists_bnd_of_pathConnected x₀ z
    rw [hz0, map_zero, sub_zero] at hv
    exact ⟨v, hv.symm⟩

/-- The partial augmentation against a set `A`: a `0`-simplex counts with
coefficient `1` when its point lies in `A` and `0` otherwise. -/
noncomputable def augFun (X : TopCat.{0}) (A : Set X) :
    Cgrp X 0 ⟶ ModuleCat.of ℤ ℤ :=
  Sigma.desc fun s => if pointOf s ∈ A then 𝟙 (ModuleCat.of ℤ ℤ) else 0

What this page does not claim

The theorem does not claim that the augmentation map is an isomorphism on homology groups of positive degree. The theorem does not claim anything about the cost function, the golden ratio, or the forcing chain. The theorem does not apply to spaces that are not path-connected.

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

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