Short complexes

This file defines the category ShortComplex C of diagrams X₁ ⟶ X₂ ⟶ X₃ such that the composition is zero.

Note: This structure ShortComplex C was first introduced in the Liquid Tensor Experiment.

structure CategoryTheory.ShortComplex (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Type (max u_1 u_3)

A short complex in a category C with zero morphisms is the datum of two composable morphisms f : X₁ ⟶ X₂ and g : X₂ ⟶ X₃ such that f ≫ g = 0.

• X₁ : C

  the first (left) object of a ShortComplex

• X₂ : C

  the second (middle) object of a ShortComplex

• X₃ : C

  the third (right) object of a ShortComplex

• f : self.X₁ ⟶ self.X₂

  the first morphism of a ShortComplex

• g : self.X₂ ⟶ self.X₃

  the second morphism of a ShortComplex

• zero : CategoryStruct.comp self.f self.g = 0

  the composition of the two given morphisms is zero

@[simp]

theorem CategoryTheory.ShortComplex.zero_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (self : ShortComplex C) {Z : C} (h : self.X₃ ⟶ Z) : CategoryStruct.comp self.f (CategoryStruct.comp self.g h) = CategoryStruct.comp 0 h

structure CategoryTheory.ShortComplex.Hom {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S₁ S₂ : ShortComplex C) : Type u_3

Morphisms of short complexes are the commutative diagrams of the obvious shape.

• τ₁ : S₁.X₁ ⟶ S₂.X₁

  the morphism on the left objects

• τ₂ : S₁.X₂ ⟶ S₂.X₂

  the morphism on the middle objects

• τ₃ : S₁.X₃ ⟶ S₂.X₃

  the morphism on the right objects

• comm₁₂ : CategoryStruct.comp self.τ₁ S₂.f = CategoryStruct.comp S₁.f self.τ₂

  the left commutative square of a morphism in ShortComplex

• comm₂₃ : CategoryStruct.comp self.τ₂ S₂.g = CategoryStruct.comp S₁.g self.τ₃

  the right commutative square of a morphism in ShortComplex

theorem CategoryTheory.ShortComplex.Hom.ext {C : Type u_1} {inst✝ : Category.{u_3, u_1} C} {inst✝¹ : Limits.HasZeroMorphisms C} {S₁ S₂ : ShortComplex C} {x y : S₁.Hom S₂} (τ₁ : x.τ₁ = y.τ₁) (τ₂ : x.τ₂ = y.τ₂) (τ₃ : x.τ₃ = y.τ₃) : x = y

theorem CategoryTheory.ShortComplex.Hom.comm₁₂_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (self : S₁.Hom S₂) {Z : C} (h : S₂.X₂ ⟶ Z) : CategoryStruct.comp self.τ₁ (CategoryStruct.comp S₂.f h) = CategoryStruct.comp S₁.f (CategoryStruct.comp self.τ₂ h)

theorem CategoryTheory.ShortComplex.Hom.comm₂₃_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (self : S₁.Hom S₂) {Z : C} (h : S₂.X₃ ⟶ Z) : CategoryStruct.comp self.τ₂ (CategoryStruct.comp S₂.g h) = CategoryStruct.comp S₁.g (CategoryStruct.comp self.τ₃ h)

def CategoryTheory.ShortComplex.Hom.id {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.Hom S

The identity morphism of a short complex.

Equations

• One or more equations did not get rendered due to their size.

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theorem CategoryTheory.ShortComplex.Hom.id_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : (id S).τ₃ = CategoryStruct.id S.X₃

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theorem CategoryTheory.ShortComplex.Hom.id_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : (id S).τ₂ = CategoryStruct.id S.X₂

@[simp]

theorem CategoryTheory.ShortComplex.Hom.id_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : (id S).τ₁ = CategoryStruct.id S.X₁

def CategoryTheory.ShortComplex.Hom.comp {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃) : S₁.Hom S₃

The composition of morphisms of short complexes.

Equations

• One or more equations did not get rendered due to their size.

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theorem CategoryTheory.ShortComplex.Hom.comp_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃) : (φ₁₂.comp φ₂₃).τ₂ = CategoryStruct.comp φ₁₂.τ₂ φ₂₃.τ₂

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theorem CategoryTheory.ShortComplex.Hom.comp_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃) : (φ₁₂.comp φ₂₃).τ₃ = CategoryStruct.comp φ₁₂.τ₃ φ₂₃.τ₃

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theorem CategoryTheory.ShortComplex.Hom.comp_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃) : (φ₁₂.comp φ₂₃).τ₁ = CategoryStruct.comp φ₁₂.τ₁ φ₂₃.τ₁

instance CategoryTheory.ShortComplex.instCategory {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Category.{u_3, max u_3 u_1} (ShortComplex C)

Equations

• CategoryTheory.ShortComplex.instCategory = CategoryTheory.Category.mk ⋯ ⋯ ⋯

theorem CategoryTheory.ShortComplex.hom_ext {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (f g : S₁ ⟶ S₂) (h₁ : f.τ₁ = g.τ₁) (h₂ : f.τ₂ = g.τ₂) (h₃ : f.τ₃ = g.τ₃) : f = g

def CategoryTheory.ShortComplex.homMk {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryStruct.comp τ₁ S₂.f = CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryStruct.comp τ₂ S₂.g = CategoryStruct.comp S₁.g τ₃) : S₁ ⟶ S₂

A constructor for morphisms in ShortComplex C when the commutativity conditions are not obvious.

Equations

• CategoryTheory.ShortComplex.homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃ = { τ₁ := τ₁, τ₂ := τ₂, τ₃ := τ₃, comm₁₂ := comm₁₂, comm₂₃ := comm₂₃ }

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theorem CategoryTheory.ShortComplex.homMk_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryStruct.comp τ₁ S₂.f = CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryStruct.comp τ₂ S₂.g = CategoryStruct.comp S₁.g τ₃) : (homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃).τ₂ = τ₂

@[simp]

theorem CategoryTheory.ShortComplex.homMk_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryStruct.comp τ₁ S₂.f = CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryStruct.comp τ₂ S₂.g = CategoryStruct.comp S₁.g τ₃) : (homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃).τ₁ = τ₁

@[simp]

theorem CategoryTheory.ShortComplex.homMk_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryStruct.comp τ₁ S₂.f = CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryStruct.comp τ₂ S₂.g = CategoryStruct.comp S₁.g τ₃) : (homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃).τ₃ = τ₃

@[simp]

theorem CategoryTheory.ShortComplex.id_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : (CategoryStruct.id S).τ₁ = CategoryStruct.id S.X₁

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theorem CategoryTheory.ShortComplex.id_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : (CategoryStruct.id S).τ₂ = CategoryStruct.id S.X₂

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theorem CategoryTheory.ShortComplex.id_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : (CategoryStruct.id S).τ₃ = CategoryStruct.id S.X₃

@[simp]

theorem CategoryTheory.ShortComplex.comp_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) : (CategoryStruct.comp φ₁₂ φ₂₃).τ₁ = CategoryStruct.comp φ₁₂.τ₁ φ₂₃.τ₁

theorem CategoryTheory.ShortComplex.comp_τ₁_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) {Z : C} (h : S₃.X₁ ⟶ Z) : CategoryStruct.comp (CategoryStruct.comp φ₁₂ φ₂₃).τ₁ h = CategoryStruct.comp φ₁₂.τ₁ (CategoryStruct.comp φ₂₃.τ₁ h)

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theorem CategoryTheory.ShortComplex.comp_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) : (CategoryStruct.comp φ₁₂ φ₂₃).τ₂ = CategoryStruct.comp φ₁₂.τ₂ φ₂₃.τ₂

theorem CategoryTheory.ShortComplex.comp_τ₂_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) {Z : C} (h : S₃.X₂ ⟶ Z) : CategoryStruct.comp (CategoryStruct.comp φ₁₂ φ₂₃).τ₂ h = CategoryStruct.comp φ₁₂.τ₂ (CategoryStruct.comp φ₂₃.τ₂ h)

@[simp]

theorem CategoryTheory.ShortComplex.comp_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) : (CategoryStruct.comp φ₁₂ φ₂₃).τ₃ = CategoryStruct.comp φ₁₂.τ₃ φ₂₃.τ₃

theorem CategoryTheory.ShortComplex.comp_τ₃_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) {Z : C} (h : S₃.X₃ ⟶ Z) : CategoryStruct.comp (CategoryStruct.comp φ₁₂ φ₂₃).τ₃ h = CategoryStruct.comp φ₁₂.τ₃ (CategoryStruct.comp φ₂₃.τ₃ h)

instance CategoryTheory.ShortComplex.instZeroHom {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} : Zero (S₁ ⟶ S₂)

Equations

• CategoryTheory.ShortComplex.instZeroHom = { zero := { τ₁ := 0, τ₂ := 0, τ₃ := 0, comm₁₂ := ⋯, comm₂₃ := ⋯ } }

@[simp]

theorem CategoryTheory.ShortComplex.zero_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S₁ S₂ : ShortComplex C) : Hom.τ₁ 0 = 0

@[simp]

theorem CategoryTheory.ShortComplex.zero_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S₁ S₂ : ShortComplex C) : Hom.τ₂ 0 = 0

@[simp]

theorem CategoryTheory.ShortComplex.zero_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S₁ S₂ : ShortComplex C) : Hom.τ₃ 0 = 0

instance CategoryTheory.ShortComplex.instHasZeroMorphisms {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Limits.HasZeroMorphisms (ShortComplex C)

Equations

• CategoryTheory.ShortComplex.instHasZeroMorphisms = CategoryTheory.Limits.HasZeroMorphisms.mk ⋯ ⋯

def CategoryTheory.ShortComplex.π₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Functor (ShortComplex C) C

The first projection functor ShortComplex C ⥤ C.

Equations

• CategoryTheory.ShortComplex.π₁ = { obj := fun (S : CategoryTheory.ShortComplex C) => S.X₁, map := fun {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y) => f.τ₁, map_id := ⋯, map_comp := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.π₁_obj {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : π₁.obj S = S.X₁

@[simp]

theorem CategoryTheory.ShortComplex.π₁_map {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {X✝ Y✝ : ShortComplex C} (f : X✝ ⟶ Y✝) : π₁.map f = f.τ₁

def CategoryTheory.ShortComplex.π₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Functor (ShortComplex C) C

The second projection functor ShortComplex C ⥤ C.

Equations

• CategoryTheory.ShortComplex.π₂ = { obj := fun (S : CategoryTheory.ShortComplex C) => S.X₂, map := fun {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y) => f.τ₂, map_id := ⋯, map_comp := ⋯ }

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theorem CategoryTheory.ShortComplex.π₂_map {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {X✝ Y✝ : ShortComplex C} (f : X✝ ⟶ Y✝) : π₂.map f = f.τ₂

@[simp]

theorem CategoryTheory.ShortComplex.π₂_obj {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : π₂.obj S = S.X₂

def CategoryTheory.ShortComplex.π₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Functor (ShortComplex C) C

The third projection functor ShortComplex C ⥤ C.

Equations

• CategoryTheory.ShortComplex.π₃ = { obj := fun (S : CategoryTheory.ShortComplex C) => S.X₃, map := fun {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y) => f.τ₃, map_id := ⋯, map_comp := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.π₃_obj {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : π₃.obj S = S.X₃

@[simp]

theorem CategoryTheory.ShortComplex.π₃_map {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {X✝ Y✝ : ShortComplex C} (f : X✝ ⟶ Y✝) : π₃.map f = f.τ₃

instance CategoryTheory.ShortComplex.preservesZeroMorphisms_π₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : π₁.PreservesZeroMorphisms

instance CategoryTheory.ShortComplex.preservesZeroMorphisms_π₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : π₂.PreservesZeroMorphisms

instance CategoryTheory.ShortComplex.preservesZeroMorphisms_π₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : π₃.PreservesZeroMorphisms

instance CategoryTheory.ShortComplex.instIsIsoτ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (f : S₁ ⟶ S₂) [IsIso f] : IsIso f.τ₁

instance CategoryTheory.ShortComplex.instIsIsoτ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (f : S₁ ⟶ S₂) [IsIso f] : IsIso f.τ₂

instance CategoryTheory.ShortComplex.instIsIsoτ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (f : S₁ ⟶ S₂) [IsIso f] : IsIso f.τ₃

def CategoryTheory.ShortComplex.π₁Toπ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : π₁ ⟶ π₂

The natural transformation π₁ ⟶ π₂ induced by S.f for all S : ShortComplex C.

Equations

• CategoryTheory.ShortComplex.π₁Toπ₂ = { app := fun (S : CategoryTheory.ShortComplex C) => S.f, naturality := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.π₁Toπ₂_app {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : π₁Toπ₂.app S = S.f

def CategoryTheory.ShortComplex.π₂Toπ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : π₂ ⟶ π₃

The natural transformation π₂ ⟶ π₃ induced by S.g for all S : ShortComplex C.

Equations

• CategoryTheory.ShortComplex.π₂Toπ₃ = { app := fun (S : CategoryTheory.ShortComplex C) => S.g, naturality := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.π₂Toπ₃_app {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : π₂Toπ₃.app S = S.g

@[simp]

theorem CategoryTheory.ShortComplex.π₁Toπ₂_comp_π₂Toπ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : CategoryStruct.comp π₁Toπ₂ π₂Toπ₃ = 0

@[simp]

theorem CategoryTheory.ShortComplex.π₁Toπ₂_comp_π₂Toπ₃_assoc {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {Z : Functor (ShortComplex C) C} (h : π₃ ⟶ Z) : CategoryStruct.comp π₁Toπ₂ (CategoryStruct.comp π₂Toπ₃ h) = CategoryStruct.comp 0 h

def CategoryTheory.ShortComplex.map {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : ShortComplex D

The short complex in D obtained by applying a functor F : C ⥤ D to a short complex in C, assuming that F preserves zero morphisms.

Equations

• S.map F = CategoryTheory.ShortComplex.mk (F.map S.f) (F.map S.g) ⋯

@[simp]

theorem CategoryTheory.ShortComplex.map_g {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : (S.map F).g = F.map S.g

@[simp]

theorem CategoryTheory.ShortComplex.map_f {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : (S.map F).f = F.map S.f

@[simp]

theorem CategoryTheory.ShortComplex.map_X₁ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : (S.map F).X₁ = F.obj S.X₁

@[simp]

theorem CategoryTheory.ShortComplex.map_X₃ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : (S.map F).X₃ = F.obj S.X₃

@[simp]

theorem CategoryTheory.ShortComplex.map_X₂ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : (S.map F).X₂ = F.obj S.X₂

def CategoryTheory.ShortComplex.mapNatTrans {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) : S.map F ⟶ S.map G

The morphism of short complexes S.map F ⟶ S.map G induced by a natural transformation F ⟶ G.

Equations

• S.mapNatTrans τ = { τ₁ := τ.app S.X₁, τ₂ := τ.app S.X₂, τ₃ := τ.app S.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.mapNatTrans_τ₂ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) : (S.mapNatTrans τ).τ₂ = τ.app S.X₂

@[simp]

theorem CategoryTheory.ShortComplex.mapNatTrans_τ₃ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) : (S.mapNatTrans τ).τ₃ = τ.app S.X₃

@[simp]

theorem CategoryTheory.ShortComplex.mapNatTrans_τ₁ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) : (S.mapNatTrans τ).τ₁ = τ.app S.X₁

def CategoryTheory.ShortComplex.mapNatIso {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G) : S.map F ≅ S.map G

The isomorphism of short complexes S.map F ≅ S.map G induced by a natural isomorphism F ≅ G.

Equations

• S.mapNatIso τ = { hom := S.mapNatTrans τ.hom, inv := S.mapNatTrans τ.inv, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.mapNatIso_inv {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G) : (S.mapNatIso τ).inv = S.mapNatTrans τ.inv

@[simp]

theorem CategoryTheory.ShortComplex.mapNatIso_hom {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] (S : ShortComplex C) [Limits.HasZeroMorphisms D] {F G : Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G) : (S.mapNatIso τ).hom = S.mapNatTrans τ.hom

def CategoryTheory.Functor.mapShortComplex {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] : Functor (ShortComplex C) (ShortComplex D)

The functor ShortComplex C ⥤ ShortComplex D induced by a functor C ⥤ D which preserves zero morphisms.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Functor.mapShortComplex_map_τ₃ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X✝ Y✝ : ShortComplex C} (φ : X✝ ⟶ Y✝) : (F.mapShortComplex.map φ).τ₃ = F.map φ.τ₃

@[simp]

theorem CategoryTheory.Functor.mapShortComplex_obj {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] (S : ShortComplex C) : F.mapShortComplex.obj S = S.map F

@[simp]

theorem CategoryTheory.Functor.mapShortComplex_map_τ₂ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X✝ Y✝ : ShortComplex C} (φ : X✝ ⟶ Y✝) : (F.mapShortComplex.map φ).τ₂ = F.map φ.τ₂

@[simp]

theorem CategoryTheory.Functor.mapShortComplex_map_τ₁ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Limits.HasZeroMorphisms C] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X✝ Y✝ : ShortComplex C} (φ : X✝ ⟶ Y✝) : (F.mapShortComplex.map φ).τ₁ = F.map φ.τ₁

def CategoryTheory.ShortComplex.isoMk {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (e₁ : S₁.X₁ ≅ S₂.X₁) (e₂ : S₁.X₂ ≅ S₂.X₂) (e₃ : S₁.X₃ ≅ S₂.X₃) (comm₁₂ : CategoryStruct.comp e₁.hom S₂.f = CategoryStruct.comp S₁.f e₂.hom := by aesop_cat) (comm₂₃ : CategoryStruct.comp e₂.hom S₂.g = CategoryStruct.comp S₁.g e₃.hom := by aesop_cat) : S₁ ≅ S₂

A constructor for isomorphisms in the category ShortComplex C

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.ShortComplex.isoMk_hom_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (e₁ : S₁.X₁ ≅ S₂.X₁) (e₂ : S₁.X₂ ≅ S₂.X₂) (e₃ : S₁.X₃ ≅ S₂.X₃) (comm₁₂ : CategoryStruct.comp e₁.hom S₂.f = CategoryStruct.comp S₁.f e₂.hom := by aesop_cat) (comm₂₃ : CategoryStruct.comp e₂.hom S₂.g = CategoryStruct.comp S₁.g e₃.hom := by aesop_cat) : (isoMk e₁ e₂ e₃ comm₁₂ comm₂₃).hom.τ₃ = e₃.hom

@[simp]

theorem CategoryTheory.ShortComplex.isoMk_hom_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (e₁ : S₁.X₁ ≅ S₂.X₁) (e₂ : S₁.X₂ ≅ S₂.X₂) (e₃ : S₁.X₃ ≅ S₂.X₃) (comm₁₂ : CategoryStruct.comp e₁.hom S₂.f = CategoryStruct.comp S₁.f e₂.hom := by aesop_cat) (comm₂₃ : CategoryStruct.comp e₂.hom S₂.g = CategoryStruct.comp S₁.g e₃.hom := by aesop_cat) : (isoMk e₁ e₂ e₃ comm₁₂ comm₂₃).hom.τ₂ = e₂.hom

@[simp]

theorem CategoryTheory.ShortComplex.isoMk_inv {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (e₁ : S₁.X₁ ≅ S₂.X₁) (e₂ : S₁.X₂ ≅ S₂.X₂) (e₃ : S₁.X₃ ≅ S₂.X₃) (comm₁₂ : CategoryStruct.comp e₁.hom S₂.f = CategoryStruct.comp S₁.f e₂.hom := by aesop_cat) (comm₂₃ : CategoryStruct.comp e₂.hom S₂.g = CategoryStruct.comp S₁.g e₃.hom := by aesop_cat) : (isoMk e₁ e₂ e₃ comm₁₂ comm₂₃).inv = homMk e₁.inv e₂.inv e₃.inv ⋯ ⋯

@[simp]

theorem CategoryTheory.ShortComplex.isoMk_hom_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (e₁ : S₁.X₁ ≅ S₂.X₁) (e₂ : S₁.X₂ ≅ S₂.X₂) (e₃ : S₁.X₃ ≅ S₂.X₃) (comm₁₂ : CategoryStruct.comp e₁.hom S₂.f = CategoryStruct.comp S₁.f e₂.hom := by aesop_cat) (comm₂₃ : CategoryStruct.comp e₂.hom S₂.g = CategoryStruct.comp S₁.g e₃.hom := by aesop_cat) : (isoMk e₁ e₂ e₃ comm₁₂ comm₂₃).hom.τ₁ = e₁.hom

theorem CategoryTheory.ShortComplex.isIso_of_isIso {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (f : S₁ ⟶ S₂) [IsIso f.τ₁] [IsIso f.τ₂] [IsIso f.τ₃] : IsIso f

def CategoryTheory.ShortComplex.op {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : ShortComplex Cᵒᵖ

The opposite ShortComplex in Cᵒᵖ associated to a short complex in C.

Equations

• S.op = CategoryTheory.ShortComplex.mk S.g.op S.f.op ⋯

@[simp]

theorem CategoryTheory.ShortComplex.op_g {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.op.g = S.f.op

@[simp]

theorem CategoryTheory.ShortComplex.op_f {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.op.f = S.g.op

@[simp]

theorem CategoryTheory.ShortComplex.op_X₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.op.X₁ = Opposite.op S.X₃

@[simp]

theorem CategoryTheory.ShortComplex.op_X₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.op.X₃ = Opposite.op S.X₁

@[simp]

theorem CategoryTheory.ShortComplex.op_X₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.op.X₂ = Opposite.op S.X₂

def CategoryTheory.ShortComplex.opMap {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (φ : S₁ ⟶ S₂) : S₂.op ⟶ S₁.op

The opposite morphism in ShortComplex Cᵒᵖ associated to a morphism in ShortComplex C

Equations

• CategoryTheory.ShortComplex.opMap φ = { τ₁ := φ.τ₃.op, τ₂ := φ.τ₂.op, τ₃ := φ.τ₁.op, comm₁₂ := ⋯, comm₂₃ := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.opMap_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (φ : S₁ ⟶ S₂) : (opMap φ).τ₂ = φ.τ₂.op

@[simp]

theorem CategoryTheory.ShortComplex.opMap_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (φ : S₁ ⟶ S₂) : (opMap φ).τ₁ = φ.τ₃.op

@[simp]

theorem CategoryTheory.ShortComplex.opMap_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex C} (φ : S₁ ⟶ S₂) : (opMap φ).τ₃ = φ.τ₁.op

@[simp]

theorem CategoryTheory.ShortComplex.opMap_id {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : opMap (CategoryStruct.id S) = CategoryStruct.id S.op

def CategoryTheory.ShortComplex.unop {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : ShortComplex C

The ShortComplex in C associated to a short complex in Cᵒᵖ.

Equations

• S.unop = CategoryTheory.ShortComplex.mk S.g.unop S.f.unop ⋯

@[simp]

theorem CategoryTheory.ShortComplex.unop_g {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : S.unop.g = S.f.unop

@[simp]

theorem CategoryTheory.ShortComplex.unop_X₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : S.unop.X₂ = Opposite.unop S.X₂

@[simp]

theorem CategoryTheory.ShortComplex.unop_X₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : S.unop.X₃ = Opposite.unop S.X₁

@[simp]

theorem CategoryTheory.ShortComplex.unop_f {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : S.unop.f = S.g.unop

@[simp]

theorem CategoryTheory.ShortComplex.unop_X₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : S.unop.X₁ = Opposite.unop S.X₃

def CategoryTheory.ShortComplex.unopMap {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex Cᵒᵖ} (φ : S₁ ⟶ S₂) : S₂.unop ⟶ S₁.unop

The morphism in ShortComplex C associated to a morphism in ShortComplex Cᵒᵖ

Equations

• CategoryTheory.ShortComplex.unopMap φ = { τ₁ := φ.τ₃.unop, τ₂ := φ.τ₂.unop, τ₃ := φ.τ₁.unop, comm₁₂ := ⋯, comm₂₃ := ⋯ }

@[simp]

theorem CategoryTheory.ShortComplex.unopMap_τ₂ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex Cᵒᵖ} (φ : S₁ ⟶ S₂) : (unopMap φ).τ₂ = φ.τ₂.unop

@[simp]

theorem CategoryTheory.ShortComplex.unopMap_τ₃ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex Cᵒᵖ} (φ : S₁ ⟶ S₂) : (unopMap φ).τ₃ = φ.τ₁.unop

@[simp]

theorem CategoryTheory.ShortComplex.unopMap_τ₁ {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {S₁ S₂ : ShortComplex Cᵒᵖ} (φ : S₁ ⟶ S₂) : (unopMap φ).τ₁ = φ.τ₃.unop

@[simp]

theorem CategoryTheory.ShortComplex.unopMap_id {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : unopMap (CategoryStruct.id S) = CategoryStruct.id S.unop

def CategoryTheory.ShortComplex.opFunctor (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Functor (ShortComplex C)ᵒᵖ (ShortComplex Cᵒᵖ)

The obvious functor (ShortComplex C)ᵒᵖ ⥤ ShortComplex Cᵒᵖ.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.ShortComplex.opFunctor_obj (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : (ShortComplex C)ᵒᵖ) : (opFunctor C).obj S = (Opposite.unop S).op

@[simp]

theorem CategoryTheory.ShortComplex.opFunctor_map (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {X✝ Y✝ : (ShortComplex C)ᵒᵖ} (φ : X✝ ⟶ Y✝) : (opFunctor C).map φ = opMap φ.unop

def CategoryTheory.ShortComplex.unopFunctor (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : Functor (ShortComplex Cᵒᵖ) (ShortComplex C)ᵒᵖ

The obvious functor ShortComplex Cᵒᵖ ⥤ (ShortComplex C)ᵒᵖ.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.ShortComplex.unopFunctor_obj (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : (unopFunctor C).obj S = Opposite.op S.unop

@[simp]

theorem CategoryTheory.ShortComplex.unopFunctor_map (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] {X✝ Y✝ : ShortComplex Cᵒᵖ} (φ : X✝ ⟶ Y✝) : (unopFunctor C).map φ = (unopMap φ).op

def CategoryTheory.ShortComplex.opEquiv (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : (ShortComplex C)ᵒᵖ ≌ ShortComplex Cᵒᵖ

The obvious equivalence of categories (ShortComplex C)ᵒᵖ ≌ ShortComplex Cᵒᵖ.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.ShortComplex.opEquiv_counitIso (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : (opEquiv C).counitIso = Iso.refl ((unopFunctor C).comp (opFunctor C))

@[simp]

theorem CategoryTheory.ShortComplex.opEquiv_unitIso (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : (opEquiv C).unitIso = Iso.refl (Functor.id (ShortComplex C)ᵒᵖ)

@[simp]

theorem CategoryTheory.ShortComplex.opEquiv_functor (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : (opEquiv C).functor = opFunctor C

@[simp]

theorem CategoryTheory.ShortComplex.opEquiv_inverse (C : Type u_1) [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] : (opEquiv C).inverse = unopFunctor C

@[reducible, inline]

abbrev CategoryTheory.ShortComplex.unopOp {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex Cᵒᵖ) : S.unop.op ≅ S

The canonical isomorphism S.unop.op ≅ S for a short complex S in Cᵒᵖ

Equations

• S.unopOp = (CategoryTheory.ShortComplex.opEquiv C).counitIso.app S

@[reducible, inline]

abbrev CategoryTheory.ShortComplex.opUnop {C : Type u_1} [Category.{u_3, u_1} C] [Limits.HasZeroMorphisms C] (S : ShortComplex C) : S.op.unop ≅ S

The canonical isomorphism S.op.unop ≅ S for a short complex S

Equations

• S.opUnop = ((CategoryTheory.ShortComplex.opEquiv C).unitIso.app (Opposite.op S)).unop