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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | import OAI.Algebra.Finitistic.Rectification.VertexModuleAssembly
import OAI.Algebra.Finitistic.Encoding.EncodingLinearity
import OAI.Algebra.Finitistic.Rectification.CanonicalArrowCompatibility
namespace OAI
noncomputable section
namespace LittleFinitistic.EncodingModuleAssembly
open QuadraticEncoding VertexDecomposition VertexModuleAssembly
universe u v w
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
variable {M : Type v} {N : Type w} [AddCommGroup M] [AddCommGroup N]
[Module k M] [Module k N] [Module (B P) M] [Module (B P) N]
[IsScalarTower k (B P) M] [IsScalarTower k (B P) N]
abbrev system := ThreeVertexAlgebra.directedSystem (composition P)
/-- Reconstruct a full module isomorphism from the three actual vertex-space
isomorphisms and the two finite arrow families. Its underlying vector-space map
is precisely the direct-sum assembly of the supplied vertex isomorphisms. -/
def equiv (E : ∀ i, Space (system P) M i ≃ₗ[k] Space (system P) N i)
(h01 : ∀ a x, E 1 (arrow (system P) (i := 0) (j := 1) (arrow01 P a)
(CanonicalArrowCompatibility.arrow01_target P a) x) =
arrow (system P) (i := 0) (j := 1) (arrow01 P a)
(CanonicalArrowCompatibility.arrow01_target P a) (E 0 x))
(h12 : ∀ b x, E 2 (arrow (system P) (i := 1) (j := 2) (arrow12 P b)
(CanonicalArrowCompatibility.arrow12_target P b) x) =
arrow (system P) (i := 1) (j := 2) (arrow12 P b)
(CanonicalArrowCompatibility.arrow12_target P b) (E 1 x)) : M ≃ₗ[B P] N :=
EncodingLinearity.linearEquiv P (assembled (system P) E)
(vertex_smul (system P) E)
(fun a => arrow_smul (system P) E (arrow01 P a)
(CanonicalArrowCompatibility.arrow01_target P a)
((ConcreteQuadraticDiagram.quadratic P).ha a) (h01 a))
(fun b => arrow_smul (system P) E (arrow12 P b)
(CanonicalArrowCompatibility.arrow12_target P b)
((ConcreteQuadraticDiagram.quadratic P).hb b) (h12 b))
end LittleFinitistic.EncodingModuleAssembly
end
end OAI
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