File size: 2,851 Bytes
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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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)
/-- Extend a finite family of actual components by zero in every other degree. -/
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)
/-- Read the same family using another finite set of support degrees. -/
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 _ _
/-- Changing finite support indices adds or removes only zero terms. The maps
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
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