Shapes of homological complexes

We define a structure ComplexShape ι for describing the shapes of homological complexes indexed by a type ι. This is intended to capture chain complexes and cochain complexes, indexed by either ℕ or ℤ, as well as more exotic examples.

Rather than insisting that the indexing type has a succ function specifying where differentials should go, inside c : ComplexShape we have c.Rel : ι → ι → Prop, and when we define HomologicalComplex we only allow nonzero differentials d i j from i to j if c.Rel i j. Further, we require that { j // c.Rel i j } and { i // c.Rel i j } are subsingletons. This means that the shape consists of some union of lines, rays, intervals, and circles.

Convenience functions c.next and c.prev provide these related elements when they exist, and return their input otherwise.

This design aims to avoid certain problems arising from dependent type theory. In particular we never have to ensure morphisms d i : X i ⟶ X (succ i) compose as expected (which would often require rewriting by equations in the indexing type). Instead such identities become separate proof obligations when verifying that a complex we've constructed is of the desired shape.

If α is an AddRightCancelSemigroup, then we define up α : ComplexShape α, the shape appropriate for cohomology, so d : X i ⟶ X j is nonzero only when j = i + 1, as well as down α : ComplexShape α, appropriate for homology, so d : X i ⟶ X j is nonzero only when i = j + 1. (Later we'll introduce CochainComplex and ChainComplex as abbreviations for HomologicalComplex with one of these shapes baked in.)

structure ComplexShape (ι : Type u_1) : Type u_1

A c : ComplexShape ι describes the shape of a chain complex, with chain groups indexed by ι. Typically ι will be ℕ, ℤ, or Fin n.

There is a relation Rel : ι → ι → Prop, and we will only allow a non-zero differential from i to j when Rel i j.

There are axioms which imply { j // c.Rel i j } and { i // c.Rel i j } are subsingletons. This means that the shape consists of some union of lines, rays, intervals, and circles.

Below we define c.next and c.prev which provide these related elements.

• Rel : ι → ι → Prop

  Nonzero differentials X i ⟶ X j shall be allowed on homological complexes when Rel i j holds.

• next_eq : ∀ {i j j' : ι}, self.Rel i j → self.Rel i j' → j = j'

  There is at most one nonzero differential from X i.

• prev_eq : ∀ {i i' j : ι}, self.Rel i j → self.Rel i' j → i = i'

  There is at most one nonzero differential to X j.

theorem ComplexShape.ext {ι : Type u_1} {x y : ComplexShape ι} (Rel : x.Rel = y.Rel) : x = y

def ComplexShape.refl (ι : Type u_2) : ComplexShape ι

The complex shape where only differentials from each X.i to itself are allowed.

This is mostly only useful so we can describe the relation of "related in k steps" below.

Equations

• ComplexShape.refl ι = { Rel := fun (i j : ι) => i = j, next_eq := ⋯, prev_eq := ⋯ }

@[simp]

theorem ComplexShape.refl_Rel (ι : Type u_2) (i j : ι) : (ComplexShape.refl ι).Rel i j = (i = j)

def ComplexShape.symm {ι : Type u_1} (c : ComplexShape ι) : ComplexShape ι

The reverse of a ComplexShape.

Equations

• c.symm = { Rel := fun (i j : ι) => c.Rel j i, next_eq := ⋯, prev_eq := ⋯ }

@[simp]

theorem ComplexShape.symm_Rel {ι : Type u_1} (c : ComplexShape ι) (i j : ι) : c.symm.Rel i j = c.Rel j i

@[simp]

theorem ComplexShape.symm_symm {ι : Type u_1} (c : ComplexShape ι) : c.symm.symm = c

theorem ComplexShape.symm_bijective {ι : Type u_1} : Function.Bijective ComplexShape.symm

def ComplexShape.trans {ι : Type u_1} (c₁ c₂ : ComplexShape ι) : ComplexShape ι

The "composition" of two ComplexShapes.

We need this to define "related in k steps" later.

Equations

• c₁.trans c₂ = { Rel := Relation.Comp c₁.Rel c₂.Rel, next_eq := ⋯, prev_eq := ⋯ }

instance ComplexShape.subsingleton_next {ι : Type u_1} (c : ComplexShape ι) (i : ι) : Subsingleton { j : ι // c.Rel i j }

Equations

• ⋯ = ⋯

instance ComplexShape.subsingleton_prev {ι : Type u_1} (c : ComplexShape ι) (j : ι) : Subsingleton { i : ι // c.Rel i j }

Equations

• ⋯ = ⋯

def ComplexShape.next {ι : Type u_1} (c : ComplexShape ι) (i : ι) : ι

An arbitrary choice of index j such that Rel i j, if such exists. Returns i otherwise.

Equations

• c.next i = if h : ∃ (j : ι), c.Rel i j then h.choose else i

def ComplexShape.prev {ι : Type u_1} (c : ComplexShape ι) (j : ι) : ι

An arbitrary choice of index i such that Rel i j, if such exists. Returns j otherwise.

Equations

• c.prev j = if h : ∃ (i : ι), c.Rel i j then h.choose else j

theorem ComplexShape.next_eq' {ι : Type u_1} (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j

theorem ComplexShape.prev_eq' {ι : Type u_1} (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i

theorem ComplexShape.next_eq_self' {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ∀ (k : ι), ¬c.Rel j k) : c.next j = j

theorem ComplexShape.prev_eq_self' {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ∀ (i : ι), ¬c.Rel i j) : c.prev j = j

theorem ComplexShape.next_eq_self {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ¬c.Rel j (c.next j)) : c.next j = j

theorem ComplexShape.prev_eq_self {ι : Type u_1} (c : ComplexShape ι) (j : ι) (hj : ¬c.Rel (c.prev j) j) : c.prev j = j

def ComplexShape.up' {α : Type u_2} [AddRightCancelSemigroup α] (a : α) : ComplexShape α

The ComplexShape allowing differentials from X i to X (i+a). (For example when a = 1, a cohomology theory indexed by ℕ or ℤ)

Equations

• ComplexShape.up' a = { Rel := fun (i j : α) => i + a = j, next_eq := ⋯, prev_eq := ⋯ }

@[simp]

theorem ComplexShape.up'_Rel {α : Type u_2} [AddRightCancelSemigroup α] (a i j : α) : (ComplexShape.up' a).Rel i j = (i + a = j)

def ComplexShape.down' {α : Type u_2} [AddRightCancelSemigroup α] (a : α) : ComplexShape α

The ComplexShape allowing differentials from X (j+a) to X j. (For example when a = 1, a homology theory indexed by ℕ or ℤ)

Equations

• ComplexShape.down' a = { Rel := fun (i j : α) => j + a = i, next_eq := ⋯, prev_eq := ⋯ }

@[simp]

theorem ComplexShape.down'_Rel {α : Type u_2} [AddRightCancelSemigroup α] (a i j : α) : (ComplexShape.down' a).Rel i j = (j + a = i)

theorem ComplexShape.down'_mk {α : Type u_2} [AddRightCancelSemigroup α] (a i j : α) (h : j + a = i) : (ComplexShape.down' a).Rel i j

def ComplexShape.up (α : Type u_2) [AddRightCancelSemigroup α] [One α] : ComplexShape α

The ComplexShape appropriate for cohomology, so d : X i ⟶ X j only when j = i + 1.

Equations

• ComplexShape.up α = ComplexShape.up' 1

@[simp]

theorem ComplexShape.up_Rel (α : Type u_2) [AddRightCancelSemigroup α] [One α] (i j : α) : (ComplexShape.up α).Rel i j = (i + 1 = j)

def ComplexShape.down (α : Type u_2) [AddRightCancelSemigroup α] [One α] : ComplexShape α

The ComplexShape appropriate for homology, so d : X i ⟶ X j only when i = j + 1.

Equations

• ComplexShape.down α = ComplexShape.down' 1

@[simp]

theorem ComplexShape.down_Rel (α : Type u_2) [AddRightCancelSemigroup α] [One α] (i j : α) : (ComplexShape.down α).Rel i j = (j + 1 = i)

theorem ComplexShape.down_mk {α : Type u_2} [AddRightCancelSemigroup α] [One α] (i j : α) (h : j + 1 = i) : (ComplexShape.down α).Rel i j

instance ComplexShape.instDecidableRelRelUp' (α : Type u_1) [AddRightCancelSemigroup α] [DecidableEq α] (a : α) : DecidableRel (ComplexShape.up' a).Rel

Equations

• ComplexShape.instDecidableRelRelUp' α a x✝ x = id inferInstance

instance ComplexShape.instDecidableRelRelDown' (α : Type u_1) [AddRightCancelSemigroup α] [DecidableEq α] (a : α) : DecidableRel (ComplexShape.down' a).Rel

Equations

• ComplexShape.instDecidableRelRelDown' α a x✝ x = id inferInstance

instance ComplexShape.instDecidableRelRelUp (α : Type u_1) [AddRightCancelSemigroup α] [DecidableEq α] [One α] : DecidableRel (ComplexShape.up α).Rel

Equations

• ComplexShape.instDecidableRelRelUp α = id inferInstance

instance ComplexShape.instDecidableRelRelDown (α : Type u_1) [AddRightCancelSemigroup α] [DecidableEq α] [One α] : DecidableRel (ComplexShape.down α).Rel

Equations

• ComplexShape.instDecidableRelRelDown α = id inferInstance