| 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 | |