Foundation Singular Mayer Vietoris Epi Homology Map Of Elementwise

The epimorphism result

The declaration epi_homologyMap_of_elementwise is a formal theorem in the framework's machine-checked library of formal theorems. It states that a particular map between homology groups is an epimorphism, meaning it is surjective: every element in the target group is the image of some element in the source group. The condition is that the map is an epimorphism elementwise, that is, at each level of the chain complex. This is a standard algebraic fact, and the framework's library proves it in the context of its singular Mayer-Vietoris construction.

The proof of this theorem uses the classical Mayer-Vietoris sequence, a tool from algebraic topology that relates the homology of a space to the homology of two open subsets that cover it. The framework constructs a sequence of abelian groups and maps, and the theorem shows that under the elementwise epimorphism condition, the induced map on homology is surjective. This is a technical but foundational result: it is the kind of lemma that allows the framework to compute homology groups in specific cases, such as when one of the covering sets is the whole space.

In Recognition Science, this theorem is part of a larger effort to build a formal foundation for topology and geometry. The framework's library contains many such lemmas, each proved from the standard axioms of mathematics. The theorem does not, by itself, establish any physical claim about the universe; it is a piece of pure mathematics. Its role is to provide a rigorous underpinning for later results that may connect to the framework's broader program.

What the theorem does not claim is also important. It does not assert that the elementwise condition holds for any particular pair of subsets. It is a conditional statement: if the elementwise epimorphism condition is satisfied, then the homology map is surjective. It also does not claim that the Mayer-Vietoris sequence itself is exact in general; that is a separate theorem in the library. The result is a building block, not a standalone conclusion about the physical world.

lemma mvSESdeg_epi (n : ℕ) : Epi (mvSESdeg U V n).g := by
  show Epi (biprod.desc (uInc U V n) (vInc U V n))
  rw [ModuleCat.epi_iff_surjective]
  intro y
  induction y using freeInduction with
  | unit t =>
      rcases t.2 with h | h
      · refine ⟨(biprod.inl : Cgrp (TopCat.of U) n ⟶ _)
          (unitOf (liftIdx U t.1 h)), ?_⟩
        rw [← ModuleCat.comp_apply, biprod.inl_desc, uInc_unitOf]
        congr 1
      · refine ⟨(biprod.inr : Cgrp (TopCat.of V) n ⟶ _)
          (unitOf (liftIdx V t.1 h)), ?_⟩
        rw [← ModuleCat.comp_apply, biprod.inr_desc, vInc_unitOf]
        congr 1
  | zero => exact ⟨0, map_zero _⟩
  | add x y hx hy =>
      obtain ⟨a, ha⟩ := hx
      obtain ⟨b, hb⟩ := hy
      exact ⟨a + b, by rw [map_add, ha, hb]⟩
  | smulz c x hx =>
      obtain ⟨a, ha⟩ := hx
      exact ⟨c • a, by rw [mapSmul, ha]⟩

lemma mvSESdeg_epi (n : ℕ) : Epi (mvSESdeg U V n).g := by
  show Epi (biprod.desc (uInc U V n) (vInc U V n))
  rw [ModuleCat.epi_iff_surjective]
  intro y
  induction y using freeInduction with
  | unit t =>
      rcases t.2 with h | h
      · refine ⟨(biprod.inl : Cgrp (TopCat.of U) n ⟶ _)
          (unitOf (liftIdx U t.1 h)), ?_⟩
        rw [← ModuleCat.comp_apply, biprod.inl_desc, uInc_unitOf]
        congr 1
      · refine ⟨(biprod.inr : Cgrp (TopCat.of V) n ⟶ _)
          (unitOf (liftIdx V t.1 h)), ?_⟩
        rw [← ModuleCat.comp_apply, biprod.inr_desc, vInc_unitOf]
        congr 1
  | zero => exact ⟨0, map_zero _⟩
  | add x y hx hy =>
      obtain ⟨a, ha⟩ := hx
      obtain ⟨b, hb⟩ := hy
      exact ⟨a + b, by rw [map_add, ha, hb]⟩
  | smulz c x hx =>
      obtain ⟨a, ha⟩ := hx
      exact ⟨c • a, by rw [mapSmul, ha]⟩

lemma mvSESdeg_exact (n : ℕ) : (mvSESdeg U V n).Exact := by
  rw [ShortComplex.moduleCat_exact_iff]
  intro z hz
  have hz' : uInc U V n
      ((biprod.fst : Cgrp (TopCat.of U) n ⊞ Cgrp (TopCat.of V) n ⟶ _) z) +
      vInc U V n
      ((biprod.snd : Cgrp (TopCat.of U) n ⊞ Cgrp (TopCat.of V) n ⟶ _) z) = 0 := by
    rw [← descApp]
    exact hz
  obtain ⟨x, hxU, hxV⟩ := mv_middle_exact U V n hz'
  refine ⟨x, ?_⟩
  apply biprod_elem_ext
  · rw [show (mvSESdeg U V n).f x = biprod.lift (chainMap (mvInclU U V) n)
      (-(chainMap (mvInclV U V) n)) x from rfl, fst_liftApp]
    exact hxU
  · rw [show (mvSESdeg U V n).f x = biprod.lift (chainMap (mvInclU U V) n)
      (-(chainMap (mvInclV U V) n)) x from rfl, snd_liftApp, negApp, hxV,
      neg_neg]

What this page does not claim

The theorem does not assert that the elementwise condition holds for any particular pair of subsets. The theorem does not claim that the Mayer-Vietoris sequence is exact in general. The theorem does not establish any physical claim about the universe.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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