Augmentation and truncation of ℕ-indexed (co)chain complexes.

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theorem ChainComplex.truncate_obj_X {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ) : (ChainComplex.truncate.toPrefunctor.obj C).X i = C.X (i + 1)

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theorem ChainComplex.truncate_obj_d {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ) (j : ℕ) : (ChainComplex.truncate.toPrefunctor.obj C).d i j = C.d (i + 1) (j + 1)

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theorem ChainComplex.truncate_map_f {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] : ∀ {X Y : ChainComplex V ℕ} (f : X ⟶ Y) (i : ℕ), (ChainComplex.truncate.toPrefunctor.map f).f i = f.f (i + 1)

def ChainComplex.truncate {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] : CategoryTheory.Functor (ChainComplex V ℕ) (ChainComplex V ℕ)

The truncation of an ℕ-indexed chain complex, deleting the object at 0 and shifting everything else down.

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def ChainComplex.truncateTo {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroObject V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) : ChainComplex.truncate.toPrefunctor.obj C ⟶ (ChainComplex.single₀ V).toPrefunctor.obj (C.X 0)

There is a canonical chain map from the truncation of a chain map C to the "single object" chain complex consisting of the truncated object C.X 0 in degree 0. The components of this chain map are C.d 1 0 in degree 0, and zero otherwise.

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def ChainComplex.augment {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) : ChainComplex V ℕ

We can "augment" a chain complex by inserting an arbitrary object in degree zero (shifting everything else up), along with a suitable differential.

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

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theorem ChainComplex.augment_X_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) : (ChainComplex.augment C f w).X 0 = X

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theorem ChainComplex.augment_X_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) (i : ℕ) : (ChainComplex.augment C f w).X (i + 1) = C.X i

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theorem ChainComplex.augment_d_one_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) : (ChainComplex.augment C f w).d 1 0 = f

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theorem ChainComplex.augment_d_succ_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) (i : ℕ) (j : ℕ) : (ChainComplex.augment C f w).d (i + 1) (j + 1) = C.d i j

def ChainComplex.truncateAugment {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) : ChainComplex.truncate.toPrefunctor.obj (ChainComplex.augment C f w) ≅ C

Truncating an augmented chain complex is isomorphic (with components the identity) to the original complex.

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theorem ChainComplex.truncateAugment_hom_f {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) (i : ℕ) : (ChainComplex.truncateAugment C f w).hom.f i = CategoryTheory.CategoryStruct.id (C.X i)

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theorem ChainComplex.truncateAugment_inv_f {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) (i : ℕ) : (ChainComplex.truncateAugment C f w).inv.f i = CategoryTheory.CategoryStruct.id ((ChainComplex.truncate.toPrefunctor.obj (ChainComplex.augment C f w)).X i)

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theorem ChainComplex.chainComplex_d_succ_succ_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ) : C.d (i + 2) 0 = 0

def ChainComplex.augmentTruncate {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) : ChainComplex.augment (ChainComplex.truncate.toPrefunctor.obj C) (C.d 1 0) (_ : CategoryTheory.CategoryStruct.comp (C.d (1 + 1) (0 + 1)) (C.d (0 + 1) 0) = 0) ≅ C

Augmenting a truncated complex with the original object and morphism is isomorphic (with components the identity) to the original complex.

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theorem ChainComplex.augmentTruncate_hom_f_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) : (ChainComplex.augmentTruncate C).hom.f 0 = CategoryTheory.CategoryStruct.id (C.X 0)

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theorem ChainComplex.augmentTruncate_hom_f_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ) : (ChainComplex.augmentTruncate C).hom.f (i + 1) = CategoryTheory.CategoryStruct.id (C.X (i + 1))

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theorem ChainComplex.augmentTruncate_inv_f_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) : (ChainComplex.augmentTruncate C).inv.f 0 = CategoryTheory.CategoryStruct.id (C.X 0)

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theorem ChainComplex.augmentTruncate_inv_f_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ) : (ChainComplex.augmentTruncate C).inv.f (i + 1) = CategoryTheory.CategoryStruct.id (C.X (i + 1))

def ChainComplex.toSingle₀AsComplex {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (C : ChainComplex V ℕ) (X : V) (f : C ⟶ (ChainComplex.single₀ V).toPrefunctor.obj X) : ChainComplex V ℕ

A chain map from a chain complex to a single object chain complex in degree zero can be reinterpreted as a chain complex.

This is the inverse construction of truncateTo.

Equations

• ChainComplex.toSingle₀AsComplex C X f = match (ChainComplex.toSingle₀Equiv C X) f with | { val := f, property := w } => ChainComplex.augment C f w

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theorem CochainComplex.truncate_obj_d {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ) (j : ℕ) : (CochainComplex.truncate.toPrefunctor.obj C).d i j = C.d (i + 1) (j + 1)

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theorem CochainComplex.truncate_obj_X {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ) : (CochainComplex.truncate.toPrefunctor.obj C).X i = C.X (i + 1)

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theorem CochainComplex.truncate_map_f {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] : ∀ {X Y : CochainComplex V ℕ} (f : X ⟶ Y) (i : ℕ), (CochainComplex.truncate.toPrefunctor.map f).f i = f.f (i + 1)

def CochainComplex.truncate {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] : CategoryTheory.Functor (CochainComplex V ℕ) (CochainComplex V ℕ)

The truncation of an ℕ-indexed cochain complex, deleting the object at 0 and shifting everything else down.

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def CochainComplex.toTruncate {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroObject V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) : (CochainComplex.single₀ V).toPrefunctor.obj (C.X 0) ⟶ CochainComplex.truncate.toPrefunctor.obj C

There is a canonical chain map from the truncation of a cochain complex C to the "single object" cochain complex consisting of the truncated object C.X 0 in degree 0. The components of this chain map are C.d 0 1 in degree 0, and zero otherwise.

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

def CochainComplex.augment {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) : CochainComplex V ℕ

We can "augment" a cochain complex by inserting an arbitrary object in degree zero (shifting everything else up), along with a suitable differential.

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

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theorem CochainComplex.augment_X_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) : (CochainComplex.augment C f w).X 0 = X

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theorem CochainComplex.augment_X_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) (i : ℕ) : (CochainComplex.augment C f w).X (i + 1) = C.X i

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theorem CochainComplex.augment_d_zero_one {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) : (CochainComplex.augment C f w).d 0 1 = f

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theorem CochainComplex.augment_d_succ_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) (i : ℕ) (j : ℕ) : (CochainComplex.augment C f w).d (i + 1) (j + 1) = C.d i j

def CochainComplex.truncateAugment {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) : CochainComplex.truncate.toPrefunctor.obj (CochainComplex.augment C f w) ≅ C

Truncating an augmented cochain complex is isomorphic (with components the identity) to the original complex.

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theorem CochainComplex.truncateAugment_hom_f {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) (i : ℕ) : (CochainComplex.truncateAugment C f w).hom.f i = CategoryTheory.CategoryStruct.id (C.X i)

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theorem CochainComplex.truncateAugment_inv_f {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : CategoryTheory.CategoryStruct.comp f (C.d 0 1) = 0) (i : ℕ) : (CochainComplex.truncateAugment C f w).inv.f i = CategoryTheory.CategoryStruct.id ((CochainComplex.truncate.toPrefunctor.obj (CochainComplex.augment C f w)).X i)

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theorem CochainComplex.cochainComplex_d_succ_succ_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ) : C.d 0 (i + 2) = 0

def CochainComplex.augmentTruncate {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) : CochainComplex.augment (CochainComplex.truncate.toPrefunctor.obj C) (C.d 0 1) (_ : CategoryTheory.CategoryStruct.comp (C.d 0 1) (C.d 1 (1 + 1)) = 0) ≅ C

Augmenting a truncated complex with the original object and morphism is isomorphic (with components the identity) to the original complex.

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

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theorem CochainComplex.augmentTruncate_hom_f_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) : (CochainComplex.augmentTruncate C).hom.f 0 = CategoryTheory.CategoryStruct.id (C.X 0)

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theorem CochainComplex.augmentTruncate_hom_f_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ) : (CochainComplex.augmentTruncate C).hom.f (i + 1) = CategoryTheory.CategoryStruct.id (C.X (i + 1))

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theorem CochainComplex.augmentTruncate_inv_f_zero {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) : (CochainComplex.augmentTruncate C).inv.f 0 = CategoryTheory.CategoryStruct.id (C.X 0)

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theorem CochainComplex.augmentTruncate_inv_f_succ {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ) : (CochainComplex.augmentTruncate C).inv.f (i + 1) = CategoryTheory.CategoryStruct.id (C.X (i + 1))

def CochainComplex.fromSingle₀AsComplex {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (C : CochainComplex V ℕ) (X : V) (f : (CochainComplex.single₀ V).toPrefunctor.obj X ⟶ C) : CochainComplex V ℕ

A chain map from a single object cochain complex in degree zero to a cochain complex can be reinterpreted as a cochain complex.

This is the inverse construction of toTruncate.

Equations

• CochainComplex.fromSingle₀AsComplex C X f = match (CochainComplex.fromSingle₀Equiv C X) f with | { val := f, property := w } => CochainComplex.augment C f w