Foundation Singular Mayer Vietoris Mv Sum Epi Of Left Univ

The epimorphism theorem

The Mayer-Vietoris sequence is a standard tool in algebraic topology. It computes the homology of a space X from the homology of two open subsets U and V that cover X. The sequence is exact, meaning the image of each map equals the kernel of the next. A key map in this sequence is the sum map, which adds a cycle from U to a cycle from V. The declaration mvSum_epi_of_left_univ proves that this sum map is an epimorphism when U is the entire space X. An epimorphism in the category of abelian groups is a surjective map, so every element in the target group is hit by the sum map.

The proof is a direct consequence of the exactness of the Mayer-Vietoris sequence. When U is the whole space, the intersection U ∩ V is just V. The sequence then degenerates in a way that makes the sum map's image equal to the entire target group. The theorem is stated for any space X, any subset V, and any degree n. It does not require U or V to be open, and it does not require U ∪ V to equal X. The only condition is that U is the full set of points of X.

This result is a small but useful piece of the framework's machine-checked library, a collection of formal theorems verified by a computer. It shows that the library can handle standard algebraic topology constructions. The theorem is a lemma in the larger project of building the singular Mayer-Vietoris sequence. It is not a statement about the physical world, and it does not depend on the framework's cost function or any other RS-specific axiom.

What the theorem does not claim is just as important. It does not say that the sum map is an epimorphism for arbitrary U and V. That would require the usual assumptions that U and V are open and cover X. It also does not say anything about the kernel of the sum map, or about the other maps in the sequence. The theorem is a single, precise statement about a special case, proved from the exactness of the sequence.

/-- **Sanity lock**: when `U = univ` (so `U` alone already covers `X`), the
Mayer-Vietoris sum map `H_n(U) ⊞ H_n(V) ⟶ H_n(X)` is an epimorphism in
every degree, because its first component is induced by the isomorphism
`univ ≃ X`. -/
theorem mvSum_epi_of_left_univ (V : Set X) (n : ℕ) :
    Epi (mvSum (Set.univ : Set X) V n) := by
  haveI hiso : IsIso (SingularPair.subInc X (Set.univ : Set X)) := by
    refine ⟨TopCat.ofHom ⟨fun x => ⟨x, trivial⟩,
      Continuous.subtype_mk continuous_id fun _ => trivial⟩, ?_, ?_⟩
    · ext x
      rfl
    · ext x
      rfl
  haveI h1 : IsIso (sChainMap (SingularPair.subInc X (Set.univ : Set X))) := by
    show IsIso (((AlgebraicTopology.singularChainComplexFunctor
      (ModuleCat.{0} ℤ)).obj (ModuleCat.of ℤ ℤ)).map
      (SingularPair.subInc X (Set.univ : Set X)))
    infer_instance
  haveI h2 : Epi (HomologicalComplex.homologyMap
      (sChainMap (SingularPair.subInc X (Set.univ : Set X))) n) := by
    haveI : IsIso (HomologicalComplex.homologyMap
        (sChainMap (SingularPair.subInc X (Set.univ : Set X))) n) := by
      show IsIso ((HF n).map (sChainMap (SingularPair.subInc X (Set.univ : Set X))))
      infer_instance
    infer_instance
  have hfac : (biprod.inl :
      (SC (TopCat.of (Set.univ : Set X))).homology n ⟶ _) ≫
      mvSum (Set.univ : Set X) V n =
      HomologicalComplex.homologyMap
        (sChainMap (SingularPair.subInc X (Set.univ : Set X))) n :=
    biprod.inl_desc _ _
  exact epi_of_epi_fac hfac

What this page does not claim

The sum map is an epimorphism for arbitrary open coverings U and V. The theorem describes any physical or empirical content. The proof depends on the framework's cost function or any RS-specific axiom.

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

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