| import Mathlib.Algebra.Category.ModuleCat.Basic |
| import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects |
| import Mathlib.LinearAlgebra.Pi |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.FiniteSupportReindex |
| universe u |
| variable {R : Type u} [Ring R] |
| variable (V : ℤ → ModuleCat.{u} R) |
| variable {l m : ℕ} (p : Fin l → ℤ) (q : Fin m → ℤ) |
|
|
| / |
| def coefficient (n : ℤ) : (∀ i, V (p i)) →ₗ[R] V n := by |
| classical |
| exact if h : ∃ i, p i = n then |
| ((eqToHom (congrArg V h.choose_spec)).hom).comp (LinearMap.proj h.choose) |
| else 0 |
|
|
| @[simp] theorem coefficient_at (hp : Function.Injective p) (i : Fin l) (x : ∀ i, V (p i)) : |
| coefficient V p (p i) x = x i := by |
| classical |
| unfold coefficient |
| split_ifs with h |
| · have transport (j : Fin l) (hji : j=i) (hpi : p j=p i) : |
| (eqToHom (congrArg V hpi)).hom (x j) = x i := by |
| subst j |
| rfl |
| exact transport h.choose (hp h.choose_spec) h.choose_spec |
| · exact False.elim (h ⟨i,rfl⟩) |
|
|
| theorem coefficient_outside (n : ℤ) (hn : ∀ i, p i ≠ n) : coefficient V p n = 0 := by |
| classical |
| exact dite_eq_right (by simpa only [not_exists] using hn) |
|
|
| / |
| def reindex : (∀ i, V (p i)) →ₗ[R] (∀ j, V (q j)) := |
| LinearMap.pi (fun j => coefficient V p (q j)) |
|
|
| @[simp] theorem reindex_apply (x : ∀ i, V (p i)) (j : Fin m) : |
| reindex V p q x j = coefficient V p (q j) x := rfl |
|
|
| theorem coefficient_reindex (hq : Function.Injective q) |
| (hV : ∀ n, (∀ j, q j ≠ n) → IsZero (V n)) |
| (n : ℤ) (x : ∀ i, V (p i)) : |
| coefficient V q n (reindex V p q x) = coefficient V p n x := by |
| classical |
| by_cases h : ∃ j, q j=n |
| · obtain ⟨j,rfl⟩ := h |
| rw [coefficient_at V q hq,reindex_apply] |
| · let := ModuleCat.subsingleton_of_isZero (hV n (by simpa only [not_exists] using h)) |
| exact Subsingleton.elim _ _ |
|
|
| / |
| are the actual coordinate inclusions/projections, including all degree transports. -/ |
| def equivalence (hp : Function.Injective p) (hq : Function.Injective q) |
| (hVp : ∀ n, (∀ i, p i ≠ n) → IsZero (V n)) |
| (hVq : ∀ n, (∀ j, q j ≠ n) → IsZero (V n)) : |
| (∀ i, V (p i)) ≃ₗ[R] (∀ j, V (q j)) where |
| __ := reindex V p q |
| invFun := reindex V q p |
| left_inv x := by |
| funext i |
| change coefficient V q (p i) (reindex V p q x) = x i |
| rw [coefficient_reindex V p q hq hVq,coefficient_at V p hp] |
| right_inv x := by |
| funext j |
| change coefficient V p (q j) (reindex V q p x) = x j |
| rw [coefficient_reindex V q p hp hVp,coefficient_at V q hq] |
| end LittleFinitistic.FiniteSupportReindex |
|
|
| end |
|
|
| end OAI |
|
|
