Yoneda reflection for a finite mesh presentation #
theorem
MagnitudeConjecture.MeshCategory.RightMeshData.yoneda_reflected_relation
{k : Type u}
[Field k]
{Q : Type v}
[Quiver Q]
[Fintype Q]
[(x y : Q) → Fintype (x ⟶ y)]
(T : RightMeshData Q)
(hP : T.FiniteContravariantRepresentables)
(z : { z : Q // z ∉ T.projective })
(x : Q)
(h : T.incomingCoefficientFiniteModule hP ↑z ⟶ T.contravariantRepresentableFiniteModule hP x)
(hh : CategoryTheory.CategoryStruct.comp (T.translationMapFinite hP z) h = 0)
:
let Y := CategoryTheory.linearYoneda k T.VertexCategory;
∑ a : IncomingArrow ↑z,
CategoryTheory.CategoryStruct.comp
(CategoryTheory.InducedCategory.homMk (T.incomingArrowHom ⟨T.tau z, (T.arrowEquiv z a.fst) a.snd⟩))
(Y.preimage (CategoryTheory.CategoryStruct.comp (T.incomingSummandInclusionFinite hP (↑z) a) h).hom.hom) = 0
Reflect the translation relation before specializing the mesh data to an algebra's standard form.