Pretriangulated Categories #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file contains the definition of pretriangulated categories and triangulated functors between them.

Implementation Notes #

We work under the assumption that pretriangulated categories are preadditive categories, but not necessarily additive categories, as is assumed in some sources.

TODO: generalise this to n-angulated categories as in https://arxiv.org/abs/1006.4592

source

@[class]

structure category_theory.pretriangulated (C : Type u) [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.has_shift C ℤ] [category_theory.preadditive C] [∀ (n : ℤ), (category_theory.shift_functor C n).additive] :

Type (max u v)

distinguished_triangles : set (category_theory.pretriangulated.triangle C)
isomorphic_distinguished : ∀ (T₁ : category_theory.pretriangulated.triangle C), (T₁ ∈ dist_triang C) → ∀ (T₂ : category_theory.pretriangulated.triangle C), (T₂ ≅ T₁) → (T₂ ∈ dist_triang C)
contractible_distinguished : ∀ (X : C), category_theory.pretriangulated.contractible_triangle X ∈ dist_triang C
distinguished_cocone_triangle : ∀ (X Y : C) (f : X ⟶ Y), ∃ (Z : C) (g : Y ⟶ Z) (h : Z ⟶ (category_theory.shift_functor C 1).obj X), category_theory.pretriangulated.triangle.mk f g h ∈ dist_triang C
rotate_distinguished_triangle : ∀ (T : category_theory.pretriangulated.triangle C), (T ∈ dist_triang C) ↔ T.rotate ∈ dist_triang C
complete_distinguished_triangle_morphism : ∀ (T₁ T₂ : category_theory.pretriangulated.triangle C), (T₁ ∈ dist_triang C) → (T₂ ∈ dist_triang C) → ∀ (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂), T₁.mor₁ ≫ b = a ≫ T₂.mor₁ → (∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (category_theory.shift_functor C 1).map a = c ≫ T₂.mor₃)

A preadditive category C with an additive shift, and a class of "distinguished triangles" relative to that shift is called pretriangulated if the following hold:

Any triangle that is isomorphic to a distinguished triangle is also distinguished.
Any triangle of the form (X,X,0,id,0,0) is distinguished.
For any morphism f : X ⟶ Y there exists a distinguished triangle of the form (X,Y,Z,f,g,h).
The triangle (X,Y,Z,f,g,h) is distinguished if and only if (Y,Z,X⟦1⟧,g,h,-f⟦1⟧) is.

Given a diagram:

      f       g       h
  X  ───> Y  ───> Z  ───> X⟦1⟧
  │       │                │
  │a      │b               │a⟦1⟧'
  V       V                V
  X' ───> Y' ───> Z' ───> X'⟦1⟧
      f'      g'      h'

where the left square commutes, and whose rows are distinguished triangles, there exists a morphism c : Z ⟶ Z' such that (a,b,c) is a triangle morphism.

See https://stacks.math.columbia.edu/tag/0145

Instances for category_theory.pretriangulated

category_theory.pretriangulated.has_sizeof_inst

source

theorem category_theory.pretriangulated.rot_of_dist_triangle (C : Type u) [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.has_shift C ℤ] [category_theory.preadditive C] [∀ (n : ℤ), (category_theory.shift_functor C n).additive] [hC : category_theory.pretriangulated C] (T : category_theory.pretriangulated.triangle C) (H : T ∈ dist_triang C) :

T.rotate ∈ dist_triang C

Given any distinguished triangle T, then we know T.rotate is also distinguished.

source

theorem category_theory.pretriangulated.inv_rot_of_dist_triangle (C : Type u) [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.has_shift C ℤ] [category_theory.preadditive C] [∀ (n : ℤ), (category_theory.shift_functor C n).additive] [hC : category_theory.pretriangulated C] (T : category_theory.pretriangulated.triangle C) (H : T ∈ dist_triang C) :

T.inv_rotate ∈ dist_triang C

Given any distinguished triangle T, then we know T.inv_rotate is also distinguished.

source

theorem category_theory.pretriangulated.comp_dist_triangle_mor_zero₁₂ (C : Type u) [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.has_shift C ℤ] [category_theory.preadditive C] [∀ (n : ℤ), (category_theory.shift_functor C n).additive] [hC : category_theory.pretriangulated C] (T : category_theory.pretriangulated.triangle C) (H : T ∈ dist_triang C) :

T.mor₁ ≫ T.mor₂ = 0

Given any distinguished triangle

      f       g       h
  X  ───> Y  ───> Z  ───> X⟦1⟧

the composition f ≫ g = 0. See https://stacks.math.columbia.edu/tag/0146

source

theorem category_theory.pretriangulated.comp_dist_triangle_mor_zero₂₃ (C : Type u) [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.has_shift C ℤ] [category_theory.preadditive C] [∀ (n : ℤ), (category_theory.shift_functor C n).additive] [hC : category_theory.pretriangulated C] (T : category_theory.pretriangulated.triangle C) (H : T ∈ dist_triang C) :

T.mor₂ ≫ T.mor₃ = 0

Given any distinguished triangle

      f       g       h
  X  ───> Y  ───> Z  ───> X⟦1⟧

the composition g ≫ h = 0. See https://stacks.math.columbia.edu/tag/0146

source

theorem category_theory.pretriangulated.comp_dist_triangle_mor_zero₃₁ (C : Type u) [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.has_shift C ℤ] [category_theory.preadditive C] [∀ (n : ℤ), (category_theory.shift_functor C n).additive] [hC : category_theory.pretriangulated C] (T : category_theory.pretriangulated.triangle C) (H : T ∈ dist_triang C) :

T.mor₃ ≫ (category_theory.shift_equiv C 1).functor.map T.mor₁ = 0

Given any distinguished triangle

      f       g       h
  X  ───> Y  ───> Z  ───> X⟦1⟧

the composition h ≫ f⟦1⟧ = 0. See https://stacks.math.columbia.edu/tag/0146