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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 43 44 45 46 47 48 | import OAI.Algebra.Finitistic.Bar.RecursiveTotalDifferential
namespace OAI
open CategoryTheory CategoryTheory.Limits
open scoped DirectSum
noncomputable section
namespace DerivedBar.RecursiveTotal
universe u
variable {D : Type u} [Ring D]
variable {I : Type} [Finite I] [DecidableEq I] (V : I → ModuleCat.{u} D)
(P : I → Prop) [DecidablePred P]
/-- Omitting zero summands is an isomorphism of ordinary direct sums. -/
def supportIso (hzero : ∀ i, ¬P i → IsZero (V i)) :
ModuleCat.of D (⨁ i, V i) ≅ ModuleCat.of D (⨁ i : {i // P i}, V i.val) where
hom := ModuleCat.ofHom (DirectSum.toModule D _ _ (fun i =>
if h : P i then DirectSum.lof D {i // P i} (fun i => V i.val) ⟨i,h⟩ else 0))
inv := ModuleCat.ofHom (DirectSum.toModule D _ _
(fun i => DirectSum.lof D I (fun i => V i) i.val))
hom_inv_id := by
apply ModuleCat.hom_ext
apply DirectSum.linearMap_ext
intro i
apply LinearMap.ext
intro v
by_cases h : P i
· simp only [ModuleCat.hom_comp, ModuleCat.hom_ofHom,
LinearMap.comp_apply, DirectSum.toModule_lof, dite_eq_left h,
ModuleCat.hom_id, LinearMap.id_apply]
· have hz : (𝟙 (V i)) = 0 := (hzero i h).eq_of_src _ _
have hv : v=0 := congrArg (fun f : V i ⟶ V i => f.hom v) hz
subst v
simp
inv_hom_id := by
apply ModuleCat.hom_ext
apply DirectSum.linearMap_ext
intro i
apply LinearMap.ext
intro v
simp only [ModuleCat.hom_comp, ModuleCat.hom_ofHom,
LinearMap.comp_apply, DirectSum.toModule_lof, dite_eq_left i.property,
ModuleCat.hom_id, LinearMap.id_apply]
end DerivedBar.RecursiveTotal
end
end OAI
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