File size: 2,786 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 | import OAI.Algebra.Finitistic.Bar.ChainSplitting
import Mathlib.Algebra.Homology.HomologicalComplexLimits
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex
open scoped DirectSum
noncomputable section
namespace DerivedBar
universe u
variable (D : Type u) [Ring D]
/-- The native algebraic direct sum, with its actual coproduct maps. -/
def moduleSumCocone (F : Discrete ℕ ⥤ ModuleCat.{u} D) : Cocone F where
pt := ModuleCat.of D (⨁ r : ℕ, F.obj ⟨r⟩)
ι := Discrete.natTrans (fun r => ModuleCat.ofHom (DirectSum.lof D ℕ (fun s => F.obj ⟨s⟩) r.as))
def moduleSumIsColimit (F : Discrete ℕ ⥤ ModuleCat.{u} D) : IsColimit (moduleSumCocone D F) where
desc s := ModuleCat.ofHom (DirectSum.toModule D ℕ s.pt (M := fun r => F.obj ⟨r⟩)
(fun r => (s.ι.app ⟨r⟩).hom))
fac s j := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro v
simp only [moduleSumCocone, Discrete.natTrans_app, ModuleCat.hom_comp,
ModuleCat.hom_ofHom, LinearMap.comp_apply, DirectSum.toModule_lof]
uniq s m hm := by
apply ModuleCat.hom_ext
apply DirectSum.linearMap_ext
intro r
apply LinearMap.ext
intro v
have h := congrArg (fun f => f.hom v) (hm ⟨r⟩)
change m.hom (DirectSum.lof D ℕ (fun s => F.obj ⟨s⟩) r v) = (s.ι.app ⟨r⟩).hom v at h
simp only [moduleSumCocone, LinearMap.comp_apply, ModuleCat.hom_ofHom,
DirectSum.toModule_lof]
exact h
/-- The canonical summand map into the already constructed chain direct sum. -/
def directSumComplexInclusion (S : ℕ → ChainComplex (ModuleCat.{u} D) ℕ) (r : ℕ) :
S r ⟶ directSumComplex S where
f n := ModuleCat.ofHom (DirectSum.lof D ℕ (fun s => (S s).X n) r)
comm' i j hij := by
change j+1=i at hij
subst i
apply ModuleCat.hom_ext
apply LinearMap.ext
intro v
simp only [directSumComplex, ChainComplex.of_d, ModuleCat.hom_comp,
ModuleCat.hom_ofHom]
change DirectSum.lmap (fun r => ((S r).d (j+1) j).hom)
(DirectSum.lof D ℕ (fun s => (S s).X (j+1)) r v) =
DirectSum.lof D ℕ (fun s => (S s).X j) r (((S r).d (j+1) j).hom v)
exact DirectSum.lmap_lof _ _ _
def directSumComplexCocone (S : ℕ → ChainComplex (ModuleCat.{u} D) ℕ) :
Cocone (Discrete.functor S) where
pt := directSumComplex S
ι := Discrete.natTrans (fun r => directSumComplexInclusion D S r.as)
/-- The native componentwise chain direct sum is the actual categorical coproduct. -/
def directSumComplexIsColimit (S : ℕ → ChainComplex (ModuleCat.{u} D) ℕ) :
IsColimit (directSumComplexCocone D S) := by
apply HomologicalComplex.isColimitOfEval
intro n
exact moduleSumIsColimit D (Discrete.functor S ⋙ HomologicalComplex.eval _ _ n)
end DerivedBar
end
end OAI
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