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/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import algebra.homology.homological_complex
/-!
# Flip a complex of complexes
For now we don't have double complexes as a distinct thing,
but we can model them as complexes of complexes.
Here we show how to flip a complex of complexes over the diagonal,
exchanging the horizontal and vertical directions.
-/
universes v u
open category_theory category_theory.limits
namespace homological_complex
variables {V : Type u} [category.{v} V] [has_zero_morphisms V]
variables {ι : Type*} {c : complex_shape ι} {ι' : Type*} {c' : complex_shape ι'}
/--
Flip a complex of complexes over the diagonal,
exchanging the horizontal and vertical directions.
-/
@[simps]
def flip_obj (C : homological_complex (homological_complex V c) c') :
homological_complex (homological_complex V c') c :=
{ X := λ i,
{ X := λ j, (C.X j).X i,
d := λ j j', (C.d j j').f i,
shape' := λ j j' w, by { rw C.shape j j' w, simp, },
d_comp_d' := λ j₁ j₂ j₃ _ _, congr_hom (C.d_comp_d j₁ j₂ j₃) i, },
d := λ i i',
{ f := λ j, (C.X j).d i i',
comm' := λ j j' h, ((C.d j j').comm i i').symm, },
shape' := λ i i' w, by { ext j, exact (C.X j).shape i i' w, } }.
variables V c c'
/-- Flipping a complex of complexes over the diagonal, as a functor. -/
@[simps]
def flip : homological_complex (homological_complex V c) c' ⥤
homological_complex (homological_complex V c') c :=
{ obj := λ C, flip_obj C,
map := λ C D f,
{ f := λ i,
{ f := λ j, (f.f j).f i,
comm' := λ j j' h, congr_hom (f.comm j j') i, }, }, }.
/-- Auxiliary definition for `homological_complex.flip_equivalence` .-/
@[simps]
def flip_equivalence_unit_iso :
𝟭 (homological_complex (homological_complex V c) c') ≅ flip V c c' ⋙ flip V c' c :=
nat_iso.of_components
(λ C,
{ hom :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, category.comp_id] }, },
inv :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, category.comp_id] }, } })
(λ X Y f, by { ext, dsimp, simp only [category.id_comp, category.comp_id], })
/-- Auxiliary definition for `homological_complex.flip_equivalence` .-/
@[simps]
def flip_equivalence_counit_iso :
flip V c' c ⋙ flip V c c' ≅ 𝟭 (homological_complex (homological_complex V c') c) :=
nat_iso.of_components
(λ C,
{ hom :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, category.comp_id] }, },
inv :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, category.comp_id] }, } })
(λ X Y f, by { ext, dsimp, simp only [category.id_comp, category.comp_id], })
/-- Flipping a complex of complexes over the diagonal, as an equivalence of categories. -/
@[simps]
def flip_equivalence :
homological_complex (homological_complex V c) c' ≌
homological_complex (homological_complex V c') c :=
{ functor := flip V c c',
inverse := flip V c' c,
unit_iso := flip_equivalence_unit_iso V c c',
counit_iso := flip_equivalence_counit_iso V c c', }
end homological_complex
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