open import Algebra.Group.Ab.Tensor
open import Algebra.Magma.Unital
open import Algebra.Group.Ab
open import Algebra.Prelude
open import Algebra.Monoid
open import Algebra.Group

open import Cat.Displayed.Univalence.Thin
open import Cat.Diagram.Equaliser.Kernel

import Algebra.Group.Cat.Base as Grp
import Algebra.Group.Ab.Hom as Ab

module Cat.Abelian.Base where

Abelian categories🔗

This module defines the sequence of properties which “work up to” abelian categories: Ab-enriched categories, pre-additive categories, pre-abelian categories, and abelian categories. Each concept builds on the last by adding a new categorical property on top of a precategory.

Ab-enriched categories🔗

An $\mathbf{Ab}$ -enriched category is one where each $\mathbf{Hom}$ set carries the structure of an Abelian group, such that the composition map is bilinear, hence extending to an Abelian group homomorphism

$\mathbf{Hom}(b, c) \otimes \mathbf{Hom}(a, b) \to \mathbf{Hom}(a, c)\text{,}$

where the term on the left is the tensor product of the corresponding $\mathbf{Hom}$ -groups. As the name implies, every such category has a canonical $\mathbf{Ab}$ -enrichment (made monoidal using $- \otimes -$ ), but we do not use the language of enriched category theory in our development of Abelian categories.

record Ab-category {o ℓ} (C : Precategory o ℓ) : Type (o ⊔ lsuc ℓ) where
  open Cat C public
  field
    Abelian-group-on-hom : ∀ A B → Abelian-group-on (Hom A B)

  _+_ : ∀ {A B} (f g : Hom A B) → Hom A B
  f + g = Abelian-group-on-hom _ _ .Abelian-group-on._*_ f g

  0m : ∀ {A B} → Hom A B
  0m = Abelian-group-on-hom _ _ .Abelian-group-on.1g

  Hom-grp : ∀ A B → Abelian-group ℓ
  Hom-grp A B = (el (Hom A B) (Hom-set A B)) , Abelian-group-on-hom A B

  field
    -- Composition is multilinear:
    ∘-linear-l
      : ∀ {A B C} (f g : Hom B C) (h : Hom A B)
      → (f ∘ h) + (g ∘ h) ≡ (f + g) ∘ h
    ∘-linear-r
      : ∀ {A B C} (f : Hom B C) (g h : Hom A B)
      → (f ∘ g) + (f ∘ h) ≡ f ∘ (g + h)

  ∘map : ∀ {A B C} → Ab.Hom (Hom-grp B C ⊗ Hom-grp A B) (Hom-grp A C)
  ∘map {A} {B} {C} =
    from-bilinear-map (Hom-grp B C) (Hom-grp A B) (Hom-grp A C)
      (record { map     = _∘_
              ; pres-*l = λ x y z → sym (∘-linear-l x y z)
              ; pres-*r = λ x y z → sym (∘-linear-r x y z)
              })

  module Hom {A B} = Abelian-group-on (Abelian-group-on-hom A B) renaming (_⁻¹ to inverse)
  open Hom
    using (zero-diff)
    renaming (_—_ to _-_)
    public

Note that from multilinearity of composition, it follows that the addition of

\mathbf{Hom}

-groups and composition¹ operations satisfy familiar algebraic identities, e.g.

0f = f0 = 0

-ab = (-a)b = a(-b)

, etc.

  ∘-zero-r : ∀ {A B C} {f : Hom B C} → f ∘ 0m {A} {B} ≡ 0m
  ∘-zero-r {f = f} =
    f ∘ 0m                     ≡⟨ Hom.intror Hom.inverser ⟩≡
    f ∘ 0m + (f ∘ 0m - f ∘ 0m) ≡⟨ Hom.associative ⟩≡
    (f ∘ 0m + f ∘ 0m) - f ∘ 0m ≡⟨ ap (_- f ∘ 0m) (∘-linear-r _ _ _) ⟩≡
    (f ∘ (0m + 0m)) - f ∘ 0m   ≡⟨ ap ((_- f ∘ 0m) ⊙ (f ∘_)) Hom.idl ⟩≡
    (f ∘ 0m) - f ∘ 0m          ≡⟨ Hom.inverser ⟩≡
    0m                         ∎

  ∘-zero-l : ∀ {A B C} {f : Hom A B} → 0m ∘ f ≡ 0m {A} {C}
  ∘-zero-l {f = f} =
    0m ∘ f                                   ≡⟨ Hom.introl Hom.inversel ⟩≡
    (Hom.inverse (0m ∘ f) + 0m ∘ f) + 0m ∘ f ≡⟨ sym Hom.associative ⟩≡
    Hom.inverse (0m ∘ f) + (0m ∘ f + 0m ∘ f) ≡⟨ ap (Hom.inverse (0m ∘ f) +_) (∘-linear-l _ _ _) ⟩≡
    Hom.inverse (0m ∘ f) + ((0m + 0m) ∘ f)   ≡⟨ ap ((Hom.inverse (0m ∘ f) +_) ⊙ (_∘ f)) Hom.idl ⟩≡
    Hom.inverse (0m ∘ f) + (0m ∘ f)          ≡⟨ Hom.inversel ⟩≡
    0m                                       ∎

  neg-∘-l
    : ∀ {A B C} {g : Hom B C} {h : Hom A B}
    → Hom.inverse (g ∘ h) ≡ Hom.inverse g ∘ h
  neg-∘-l {g = g} {h} = monoid-inverse-unique Hom.has-is-monoid (g ∘ h) _ _
    Hom.inversel
    (∘-linear-l _ _ _ ∙ ap (_∘ h) Hom.inverser ∙ ∘-zero-l)

  neg-∘-r
    : ∀ {A B C} {g : Hom B C} {h : Hom A B}
    → Hom.inverse (g ∘ h) ≡ g ∘ Hom.inverse h
  neg-∘-r {g = g} {h} = monoid-inverse-unique Hom.has-is-monoid (g ∘ h) _ _
    Hom.inversel
    (∘-linear-r _ _ _ ∙ ap (g ∘_) Hom.inverser ∙ ∘-zero-r)

  ∘-minus-l
    : ∀ {A B C} (f g : Hom B C) (h : Hom A B)
    → (f ∘ h) - (g ∘ h) ≡ (f - g) ∘ h
  ∘-minus-l f g h =
    f ∘ h - g ∘ h               ≡⟨ ap (f ∘ h +_) neg-∘-l ⟩≡
    f ∘ h + (Hom.inverse g ∘ h) ≡⟨ ∘-linear-l _ _ _ ⟩≡
    (f - g) ∘ h                 ∎

  ∘-minus-r
    : ∀ {A B C} (f : Hom B C) (g h : Hom A B)
    → (f ∘ g) - (f ∘ h) ≡ f ∘ (g - h)
  ∘-minus-r f g h =
    f ∘ g - f ∘ h               ≡⟨ ap (f ∘ g +_) neg-∘-r ⟩≡
    f ∘ g + (f ∘ Hom.inverse h) ≡⟨ ∘-linear-r _ _ _ ⟩≡
    f ∘ (g - h)                 ∎

Before moving on, we note the following property of $\mathbf{Ab}$ -categories: If $A$ is an object s.t. $\mathrm{id}_{A} = 0$ , then $A$ is a zero object.

module _ {o ℓ} {C : Precategory o ℓ} (A : Ab-category C) where
  private module A = Ab-category A

  id-zero→zero : ∀ {A} → A.id {A} ≡ A.0m → A.is-zero A
  id-zero→zero idm .A.is-zero.has-is-initial B = contr A.0m λ h → sym $
    h                                ≡⟨ A.intror refl ⟩≡
    h A.∘ A.id                       ≡⟨ A.refl⟩∘⟨ idm ⟩≡
    h A.∘ A.0m                       ≡⟨ A.∘-zero-r ⟩≡
    A.0m                             ∎
  id-zero→zero idm .A.is-zero.has-is-terminal x = contr A.0m λ h → sym $
    h                              ≡⟨ A.introl refl ⟩≡
    A.id A.∘ h                     ≡⟨ idm A.⟩∘⟨refl ⟩≡
    A.0m A.∘ h                     ≡⟨ A.∘-zero-l ⟩≡
    A.0m                           ∎

Perhaps the simplest example of an $\mathbf{Ab}$ -category is.. any ring! In the same way that a monoid is a category with one object, and a group is a groupoid with one object, a ring is a ringoid with one object; Ringoid being another word for $\mathbf{Ab}$ -category, rather than a horizontal categorification of the drummer for the Beatles. The next simplest example is $\mathbf{Ab}$ itself:

module _ where
  open Ab-category
  Ab-ab-category : ∀ {ℓ} → Ab-category (Ab ℓ)
  Ab-ab-category .Abelian-group-on-hom A B = Ab.Abelian-group-on-hom A B
  Ab-ab-category .∘-linear-l f g h = Homomorphism-path (λ _ → refl)
  Ab-ab-category .∘-linear-r f g h =
    Homomorphism-path (λ _ → sym (f .preserves .is-group-hom.pres-⋆ _ _))

Additive categories🔗

An $\mathbf{Ab}$ -category is additive when its underlying category has a terminal object and finite products; By the yoga above, this implies that the terminal object is also a zero object, and the finite products coincide with finite coproducts.

record is-additive {o ℓ} (C : Precategory o ℓ) : Type (o ⊔ lsuc ℓ) where
  field has-ab : Ab-category C
  open Ab-category has-ab public

  field
    has-terminal : Terminal
    has-prods    : ∀ A B → Product A B

  ∅ : Zero
  ∅ .Zero.∅ = has-terminal .Terminal.top
  ∅ .Zero.has-is-zero = id-zero→zero has-ab $
    is-contr→is-prop (has-terminal .Terminal.has⊤ _) _ _
  module ∅ = Zero ∅

  0m-unique : ∀ {A B} → ∅.zero→ {A} {B} ≡ 0m
  0m-unique = ap₂ _∘_ (∅.has⊥ _ .paths _) refl ∙ ∘-zero-l

Coincidence of finite products and finite coproducts leads to an object commonly called a (finite) biproduct. The coproduct coprojections are given by the pair of maps

$\begin{align*} &(\mathrm{id}_{} \times 0) : A \to A \times B \\ &(0 \times \mathrm{id}_{}) : B \to A \times B\text{,} \end{align*}$

respectively, and the comultiplication of $f$ and $g$ is given by $f\pi_1 + g\pi_2$ . We can calculate, for the first coprojection followed by comultiplication,

$\begin{align*} & (f\pi_1+g\pi_2)(\mathrm{id}_{}\times 0) \\ =& f\pi_1(\mathrm{id}_{}\times 0) + g\pi_2(\mathrm{id}_{}\times 0) \\ =& f\mathrm{id}_{} + g0 \\ =& f\text{,} \end{align*}$

and analogously for the second coprojection followed by comultiplication.

  has-coprods : ∀ A B → Coproduct A B
  has-coprods A B = coprod where
    open Coproduct
    open is-coproduct
    module Prod = Product (has-prods A B)
    coprod : Coproduct A B
    coprod .coapex = Prod.apex
    coprod .in₀ = Prod.⟨ id , 0m ⟩Prod.
    coprod .in₁ = Prod.⟨ 0m , id ⟩Prod.
    coprod .has-is-coproduct .[_,_] f g = f ∘ Prod.π₁ + g ∘ Prod.π₂
    coprod .has-is-coproduct .in₀∘factor {inj0 = inj0} {inj1} =
      (inj0 ∘ Prod.π₁ + inj1 ∘ Prod.π₂) ∘ Prod.⟨ id , 0m ⟩Prod. ≡⟨ sym (∘-linear-l _ _ _) ⟩≡
      ((inj0 ∘ Prod.π₁) ∘ Prod.⟨ id , 0m ⟩Prod. + _)            ≡⟨ Hom.elimr (pullr Prod.π₂∘factor ∙ ∘-zero-r) ⟩≡
      (inj0 ∘ Prod.π₁) ∘ Prod.⟨ id , 0m ⟩Prod.                  ≡⟨ cancelr Prod.π₁∘factor ⟩≡
      inj0                                                ∎
    coprod .has-is-coproduct .in₁∘factor {inj0 = inj0} {inj1} =
      (inj0 ∘ Prod.π₁ + inj1 ∘ Prod.π₂) ∘ Prod.⟨ 0m , id ⟩Prod. ≡⟨ sym (∘-linear-l _ _ _) ⟩≡
      (_ + (inj1 ∘ Prod.π₂) ∘ Prod.⟨ 0m , id ⟩Prod.)            ≡⟨ Hom.eliml (pullr Prod.π₁∘factor ∙ ∘-zero-r) ⟩≡
      (inj1 ∘ Prod.π₂) ∘ Prod.⟨ 0m , id ⟩Prod.                  ≡⟨ cancelr Prod.π₂∘factor ⟩≡
      inj1                                                 ∎

For uniqueness, we use distributivity of composition over addition of morphisms and the universal property of the product to establish the desired equation. Check it out:

    coprod .has-is-coproduct .unique {inj0 = inj0} {inj1} other p q = sym $
      inj0 ∘ Prod.π₁ + inj1 ∘ Prod.π₂                                             ≡⟨ ap₂ _+_ (pushl (sym p)) (pushl (sym q)) ⟩≡
      (other ∘ Prod.⟨ id , 0m ⟩Prod. ∘ Prod.π₁) + (other ∘ Prod.⟨ 0m , id ⟩Prod. ∘ Prod.π₂) ≡⟨ ∘-linear-r _ _ _ ⟩≡
      other ∘ (Prod.⟨ id , 0m ⟩Prod. ∘ Prod.π₁ + Prod.⟨ 0m , id ⟩Prod. ∘ Prod.π₂)           ≡⟨ elimr lemma ⟩≡
      other                                                                       ∎
      where
        lemma : Prod.⟨ id , 0m ⟩Prod. ∘ Prod.π₁ + Prod.⟨ 0m , id ⟩Prod. ∘ Prod.π₂
              ≡ id
        lemma = Prod.unique₂ {pr1 = Prod.π₁} {pr2 = Prod.π₂}
          (sym (∘-linear-r _ _ _) ∙ ap₂ _+_ (cancell Prod.π₁∘factor) (pulll Prod.π₁∘factor ∙ ∘-zero-l) ∙ Hom.elimr refl)
          (sym (∘-linear-r _ _ _) ∙ ap₂ _+_ (pulll Prod.π₂∘factor ∙ ∘-zero-l) (cancell Prod.π₂∘factor) ∙ Hom.eliml refl)
          (elimr refl)
          (elimr refl)

Pre-abelian & abelian categories🔗

An additive category is pre-abelian when it additionally has kernels and cokernels, hence binary equalisers and coequalisers where one of the maps is zero.

record is-pre-abelian {o ℓ} (C : Precategory o ℓ) : Type (o ⊔ lsuc ℓ) where
  field has-additive : is-additive C
  open is-additive has-additive public
  field
    kernel   : ∀ {A B} (f : Hom A B) → Kernel C ∅ f
    cokernel : ∀ {A B} (f : Hom A B) → Coequaliser 0m f

  module Ker {A B} (f : Hom A B) = Kernel (kernel f)
  module Coker {A B} (f : Hom A B) = Coequaliser (cokernel f)

Every morphism $A \xrightarrow{f} B$ in a preabelian category admits a canonical decomposition as

$A \xtwoheadrightarrow{p} \operatorname{coker}(\ker f) \xrightarrow{f'} \ker (\operatorname{coker}f) \xhookrightarrow{i} B\text{,}$

where, as indicated, the map $p$ is an epimorphism (indeed a regular epimorphism, since it is a cokernel) and the map $i$ is a regular monomorphism.

  decompose
    : ∀ {A B} (f : Hom A B)
    → Σ[ f′ ∈ Hom (Coker.coapex (Ker.kernel f)) (Ker.ker (Coker.coeq f)) ]
       (f ≡ Ker.kernel (Coker.coeq f) ∘ f′ ∘ Coker.coeq (Ker.kernel f))
  decompose {A} {B} f = map , sym path
    where
      proj′ : Hom (Coker.coapex (Ker.kernel f)) B
      proj′ = Coker.universal (Ker.kernel f) {e′ = f} $ sym path

      map : Hom (Coker.coapex (Ker.kernel f)) (Ker.ker (Coker.coeq f))
      map = Ker.universal (Coker.coeq f) {e′ = proj′} $ sym path

The existence of the map $f'$ , and indeed of the maps $p$ and $i$ , follow from the universal properties of kernels and cokernels. The map $p$ is the canonical quotient map $A \to \operatorname{coker}(f)$ , and the map $i$ is the canonical subobject inclusion $\ker(f) \to B$ .

A pre-abelian category is abelian when the map $f'$ in the above decomposition is an isomorphism.

record is-abelian {o ℓ} (C : Precategory o ℓ) : Type (o ⊔ lsuc ℓ) where
  field has-is-preab : is-pre-abelian C
  open is-pre-abelian has-is-preab public
  field
    coker-ker≃ker-coker
      : ∀ {A B} (f : Hom A B) → is-invertible (decompose f .fst)

This implies in particular that any monomorphism is a kernel, and every epimorphism is a cokernel. Let’s investigate the case for “every mono is a kernel” first: Suppose that $f : A \hookrightarrow B$ is some monomorphism; We’ll show that it’s isomorphic to $\ker (\operatorname{coker}f)$ in the slice category $\mathcal{A}/B$ .

  module _ {A B} (f : Hom A B) (monic : is-monic f) where
    private
      module m = Cat (Slice C B)

The map $A \to \ker (\operatorname{coker}f)$ is obtained as the composite

$A \xtwoheadrightarrow{p} \operatorname{coker}(\ker f) \cong \ker (\operatorname{coker}f)\text{,}$

where the isomorphism is our canonical map from before.

      f→kercoker : m.Hom (cut f) (cut (Ker.kernel (Coker.coeq f)))
      f→kercoker ./-Hom.map = decompose f .fst ∘ Coker.coeq (Ker.kernel f)
      f→kercoker ./-Hom.commutes = sym (decompose f .snd)

Conversely, map $\ker (\operatorname{coker}f) \to A$ is the composite

$\ker (\operatorname{coker}f) \cong \operatorname{coker}(\ker f) \to A\text{,}$

where the second map arises from the universal property of the cokernel: We can map out of it with the map $\ker f \hookrightarrow A$ , since (using that $f$ is mono), we have $0 = \ker f$ from $f0 = f\ker f$ .

      kercoker→f : m.Hom (cut (Ker.kernel (Coker.coeq f))) (cut f)
      kercoker→f ./-Hom.map =
        Coker.universal (Ker.kernel f) {e′ = id} (monic _ _ path) ∘
          coker-ker≃ker-coker f .is-invertible.inv
        where abstract
          path : f ∘ id ∘ 0m ≡ f ∘ id ∘ Ker.kernel f
          path =
            f ∘ id ∘ 0m              ≡⟨ ap (f ∘_) (eliml refl) ∙ ∘-zero-r ⟩≡
            0m                       ≡˘⟨ ∅.zero-∘r _ ∙ 0m-unique ⟩≡˘
            (∅.zero→ ∘ Ker.kernel f) ≡˘⟨ Ker.equal f ⟩≡˘
            f ∘ Ker.kernel f         ≡⟨ ap (f ∘_) (introl refl) ⟩≡
            f ∘ id ∘ Ker.kernel f    ∎

This is indeed a map in the slice using that both isomorphisms and coequalisers are epic to make progress.

      kercoker→f ./-Hom.commutes = path where
        lemma =
          is-coequaliser→is-epic (Coker.coeq _) (Coker.has-is-coeq _) _ _ $
               pullr (Coker.factors _)
            ·· elimr refl
            ·· (decompose f .snd ∙ assoc _ _ _)

        path =
          invertible→epic (coker-ker≃ker-coker _) _ _ $
            (f ∘ Coker.universal _ _ ∘ _) ∘ decompose f .fst   ≡⟨ ap₂ _∘_ (assoc _ _ _) refl ⟩≡
            ((f ∘ Coker.universal _ _) ∘ _) ∘ decompose f .fst ≡⟨ cancelr (coker-ker≃ker-coker _ .is-invertible.invr) ⟩≡
            f ∘ Coker.universal _ _                            ≡⟨ lemma ⟩≡
            Ker.kernel _ ∘ decompose f .fst                     ∎

Using the universal property of the cokernel (both uniqueness and universality), we establish that the maps defined above are inverses in $\mathcal{A}$ , thus assemble into an isomorphism in the slice.

    mono→kernel : cut f m.≅ cut (Ker.kernel (Coker.coeq f))
    mono→kernel = m.make-iso f→kercoker kercoker→f f→kc→f kc→f→kc where
      f→kc→f : f→kercoker m.∘ kercoker→f ≡ m.id
      f→kc→f = /-Hom-path $
        (decompose f .fst ∘ Coker.coeq _) ∘ Coker.universal _ _ ∘ _  ≡⟨ cancel-inner lemma ⟩≡
        decompose f .fst ∘ _                                         ≡⟨ coker-ker≃ker-coker f .is-invertible.invl ⟩≡
        id                                                           ∎
        where
          lemma = Coker.unique₂ _
            {e′ = Coker.coeq (Ker.kernel f)}
            (∘-zero-r ∙ sym (sym (Coker.coequal _) ∙ ∘-zero-r))
            (pullr (Coker.factors (Ker.kernel f)) ∙ elimr refl)
            (eliml refl)

      kc→f→kc : kercoker→f m.∘ f→kercoker ≡ m.id
      kc→f→kc = /-Hom-path $
        (Coker.universal _ _ ∘ _) ∘ decompose f .fst ∘ Coker.coeq _ ≡⟨ cancel-inner (coker-ker≃ker-coker f .is-invertible.invr) ⟩≡
        Coker.universal _ _ ∘ Coker.coeq _                          ≡⟨ Coker.factors _ ⟩≡
        id                                                           ∎

“multiplication”↩︎