| import OAI.Algebra.Finitistic.Ordinary.OrdinaryFiniteTotal | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| namespace LittleFinitistic.FiniteColumnComplex | |
| universe u | |
| variable {R : Type u} [Ring R] {l : ℕ} | |
| variable (K L : Fin l → CochainComplex (ModuleCat.{u} R) ℤ) | |
| variable (δ : Differential K) (ε : Differential L) | |
| variable (f : ∀ i, K i ⟶ L i) | |
| /-- The degreewise map between the actual finite sums of columns. -/ | |
| def componentMap (n : ℤ) : term K n →ₗ[R] term L n := | |
| LinearMap.pi (fun i => ((f i).f n).hom.comp (LinearMap.proj i)) | |
| @[simp] theorem componentMap_apply (n : ℤ) (z : term K n) (i : Fin l) : | |
| componentMap K L f n z i = ((f i).f n).hom (z i) := rfl | |
| /-- A compatible family of column maps gives an actual map of total complexes. -/ | |
| def complexMap (hc : ∀ n, | |
| (ε.d n).comp (componentMap K L f n) = (componentMap K L f (n+1)).comp (δ.d n)) : | |
| complex K δ ⟶ complex L ε where | |
| f n := ModuleCat.ofHom (componentMap K L f n) | |
| comm' n m h := by | |
| change n+1=m at h | |
| subst m | |
| rw [complex_d,complex_d] | |
| apply ModuleCat.hom_ext | |
| exact hc n | |
| end LittleFinitistic.FiniteColumnComplex | |
| end | |
| end OAI | |