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

open import Cat.Diagram.Equaliser.Kernel
open import Cat.Diagram.Coequaliser
open import Cat.Diagram.Coproduct
open import Cat.Diagram.Terminal
open import Cat.Diagram.Product
open import Cat.Diagram.Zero

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

module Cat.Abelian.Base where


# Abelian categoriesπ

This module defines the sequence of properties which βwork up toβ 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 categoriesπ

An category is one where each set carries the structure of an Abelian group, such that the composition map is bilinear, hence extending to an Abelian group homomorphism

where the term on the left is the tensor product of the corresponding As the name implies, every such category has a canonical (made monoidal using but we do not use the language of enriched category theory in our development of Abelian categories.

record Ab-category {o β} (C : Precategory o β) : Type (o β lsuc β) where
open Cat C public
field
Abelian-group-on-hom : β A B β Abelian-group-on (Hom A B)

_+_ : β {A B} (f g : Hom A B) β Hom A B
f + g = Abelian-group-on-hom _ _ .Abelian-group-on._*_ f g

0m : β {A B} β Hom A B
0m = Abelian-group-on-hom _ _ .Abelian-group-on.1g

Hom-grp : β A B β Abelian-group β
Hom-grp A B = (el (Hom A B) (Hom-set A B)) , Abelian-group-on-hom A B

field
-- Composition is multilinear:
β-linear-l
: β {A B C} (f g : Hom B C) (h : Hom A B)
β (f β h) + (g β h) β‘ (f + g) β h
β-linear-r
: β {A B C} (f : Hom B C) (g h : Hom A B)
β (f β g) + (f β h) β‘ f β (g + h)

βmap : β {A B C} β Ab.Hom (Hom-grp B C β Hom-grp A B) (Hom-grp A C)
βmap {A} {B} {C} =
from-bilinear-map (Hom-grp B C) (Hom-grp A B) (Hom-grp A C)
(record { map     = _β_
; pres-*l = Ξ» x y z β sym (β-linear-l x y z)
; pres-*r = Ξ» x y z β sym (β-linear-r x y z)
})

module Hom {A B} = Abelian-group-on (Abelian-group-on-hom A B) renaming (_β»ΒΉ to inverse)
open Hom
using (zero-diff)
renaming (_β_ to _-_)
public

Note that from multilinearity of composition, it follows that the addition of and composition1 operations satisfy familiar algebraic identities, e.g.Β  etc.
  β-zero-r : β {A B C} {f : Hom B C} β f β 0m {A} {B} β‘ 0m
β-zero-r {f = f} =
f β 0m                     β‘β¨ Hom.intror Hom.inverser β©β‘
f β 0m + (f β 0m - f β 0m) β‘β¨ Hom.associative β©β‘
(f β 0m + f β 0m) - f β 0m β‘β¨ ap (_- f β 0m) (β-linear-r _ _ _) β©β‘
(f β (0m + 0m)) - f β 0m   β‘β¨ ap ((_- f β 0m) β (f β_)) Hom.idl β©β‘
(f β 0m) - f β 0m          β‘β¨ Hom.inverser β©β‘
0m                         β

β-zero-l : β {A B C} {f : Hom A B} β 0m β f β‘ 0m {A} {C}
β-zero-l {f = f} =
0m β f                                   β‘β¨ Hom.introl Hom.inversel β©β‘
(Hom.inverse (0m β f) + 0m β f) + 0m β f β‘β¨ sym Hom.associative β©β‘
Hom.inverse (0m β f) + (0m β f + 0m β f) β‘β¨ ap (Hom.inverse (0m β f) +_) (β-linear-l _ _ _) β©β‘
Hom.inverse (0m β f) + ((0m + 0m) β f)   β‘β¨ ap ((Hom.inverse (0m β f) +_) β (_β f)) Hom.idl β©β‘
Hom.inverse (0m β f) + (0m β f)          β‘β¨ Hom.inversel β©β‘
0m                                       β

neg-β-l
: β {A B C} {g : Hom B C} {h : Hom A B}
β Hom.inverse (g β h) β‘ Hom.inverse g β h
neg-β-l {g = g} {h} = monoid-inverse-unique Hom.has-is-monoid (g β h) _ _
Hom.inversel
(β-linear-l _ _ _ β ap (_β h) Hom.inverser β β-zero-l)

neg-β-r
: β {A B C} {g : Hom B C} {h : Hom A B}
β Hom.inverse (g β h) β‘ g β Hom.inverse h
neg-β-r {g = g} {h} = monoid-inverse-unique Hom.has-is-monoid (g β h) _ _
Hom.inversel
(β-linear-r _ _ _ β ap (g β_) Hom.inverser β β-zero-r)

β-minus-l
: β {A B C} (f g : Hom B C) (h : Hom A B)
β (f β h) - (g β h) β‘ (f - g) β h
β-minus-l f g h =
f β h - g β h               β‘β¨ ap (f β h +_) neg-β-l β©β‘
f β h + (Hom.inverse g β h) β‘β¨ β-linear-l _ _ _ β©β‘
(f - g) β h                 β

β-minus-r
: β {A B C} (f : Hom B C) (g h : Hom A B)
β (f β g) - (f β h) β‘ f β (g - h)
β-minus-r f g h =
f β g - f β h               β‘β¨ ap (f β g +_) neg-β-r β©β‘
f β g + (f β Hom.inverse h) β‘β¨ β-linear-r _ _ _ β©β‘
f β (g - h)                 β


Before moving on, we note the following property of If is an object s.t. then is a zero object.

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

id-zeroβzero : β {X} β A.id {X} β‘ A.0m β is-zero C X
id-zeroβzero idm .is-zero.has-is-initial B = contr A.0m Ξ» h β sym $h β‘β¨ A.intror refl β©β‘ h A.β A.id β‘β¨ A.reflβ©ββ¨ idm β©β‘ h A.β A.0m β‘β¨ A.β-zero-r β©β‘ A.0m β id-zeroβzero idm .is-zero.has-is-terminal x = contr A.0m Ξ» h β sym$
A.0m A.β h                     β‘β¨ A.β-zero-l β©β‘
A.0m                           β


Perhaps the simplest example of an 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 rather than a horizontal categorification of the drummer for the Beatles. The next simplest example is itself:

module _ where
open Ab-category
Ab-ab-category : β {β} β Ab-category (Ab β)
Ab-ab-category .Abelian-group-on-hom A B = Ab.Abelian-group-on-hom A B
Ab-ab-category .β-linear-l f g h = trivial!
Ab-ab-category .β-linear-r f g h = ext Ξ» _ β
sym (f .preserves .is-group-hom.pres-β _ _)


An 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 β} (C : Precategory o β) : Type (o β lsuc β) where
field has-ab : Ab-category C
open Ab-category has-ab public

field
has-terminal : Terminal C
has-prods    : β A B β Product C A B

β : Zero C
β .Zero.β = has-terminal .Terminal.top
β .Zero.has-is-zero = id-zeroβzero has-ab $is-contrβis-prop (has-terminal .Terminal.hasβ€ _) _ _ module β = Zero β 0m-unique : β {A B} β β .zeroβ {A} {B} β‘ 0m 0m-unique = apβ _β_ (β .hasβ₯ _ .paths _) refl β β-zero-l  Coincidence of finite products and finite coproducts leads to an object commonly called a (finite) biproduct. The coproduct coprojections are given by the pair of maps respectively, and the comultiplication of and is given by We can calculate, for the first coprojection followed by comultiplication, and analogously for the second coprojection followed by comultiplication.  has-coprods : β A B β Coproduct C A B has-coprods A B = coprod where open Coproduct open is-coproduct module Prod = Product (has-prods A B) coprod : Coproduct C A B coprod .coapex = Prod.apex coprod .inβ = Prod.β¨ id , 0m β©Prod. coprod .inβ = Prod.β¨ 0m , id β©Prod. coprod .has-is-coproduct .[_,_] f g = f β Prod.Οβ + g β Prod.Οβ coprod .has-is-coproduct .inββfactor {inj0 = inj0} {inj1} = (inj0 β Prod.Οβ + inj1 β Prod.Οβ) β Prod.β¨ id , 0m β©Prod. β‘β¨ sym (β-linear-l _ _ _) β©β‘ ((inj0 β Prod.Οβ) β Prod.β¨ id , 0m β©Prod. + _) β‘β¨ Hom.elimr (pullr Prod.Οββfactor β β-zero-r) β©β‘ (inj0 β Prod.Οβ) β Prod.β¨ id , 0m β©Prod. β‘β¨ cancelr Prod.Οββfactor β©β‘ inj0 β coprod .has-is-coproduct .inββfactor {inj0 = inj0} {inj1} = (inj0 β Prod.Οβ + inj1 β Prod.Οβ) β Prod.β¨ 0m , id β©Prod. β‘β¨ sym (β-linear-l _ _ _) β©β‘ (_ + (inj1 β Prod.Οβ) β Prod.β¨ 0m , id β©Prod.) β‘β¨ Hom.eliml (pullr Prod.Οββfactor β β-zero-r) β©β‘ (inj1 β Prod.Οβ) β Prod.β¨ 0m , id β©Prod. β‘β¨ cancelr Prod.Οββfactor β©β‘ inj1 β  For uniqueness, we use distributivity of composition over addition of morphisms and the universal property of the product to establish the desired equation. Check it out:  coprod .has-is-coproduct .unique {inj0 = inj0} {inj1} other p q = sym$
inj0 β Prod.Οβ + inj1 β Prod.Οβ                                             β‘β¨ apβ _+_ (pushl (sym p)) (pushl (sym q)) β©β‘
(other β Prod.β¨ id , 0m β©Prod. β Prod.Οβ) + (other β Prod.β¨ 0m , id β©Prod. β Prod.Οβ) β‘β¨ β-linear-r _ _ _ β©β‘
other β (Prod.β¨ id , 0m β©Prod. β Prod.Οβ + Prod.β¨ 0m , id β©Prod. β Prod.Οβ)           β‘β¨ elimr lemma β©β‘
other                                                                       β
where
lemma : Prod.β¨ id , 0m β©Prod. β Prod.Οβ + Prod.β¨ 0m , id β©Prod. β Prod.Οβ
β‘ id
lemma = Prod.uniqueβ {pr1 = Prod.Οβ} {pr2 = Prod.Οβ}
(sym (β-linear-r _ _ _) β apβ _+_ (cancell Prod.Οββfactor) (pulll Prod.Οββfactor β β-zero-l) β Hom.elimr refl)
(sym (β-linear-r _ _ _) β apβ _+_ (pulll Prod.Οββfactor β β-zero-l) (cancell Prod.Οββfactor) β Hom.eliml refl)
(elimr refl)
(elimr refl)


# Pre-abelian & abelian categoriesπ

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 β} (C : Precategory o β) : Type (o β lsuc β) where
field
kernel   : β {A B} (f : Hom A B) β Kernel C β f
cokernel : β {A B} (f : Hom A B) β Coequaliser C 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 in a preabelian category admits a canonical decomposition as

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

  decompose
: β {A B} (f : Hom A B)
β Ξ£[ f' β Hom (Coker.coapex (Ker.kernel f)) (Ker.ker (Coker.coeq f)) ]
(f β‘ Ker.kernel (Coker.coeq f) β f' β 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} $sym path   where abstract path : f β kernel f .Kernel.kernel β‘ f β 0m path = Ker.equal f Β·Β· β .zero-βr _ Β·Β· apβ _β_ (β .hasβ₯ _ .paths 0m) refl Β·Β· β-zero-l Β·Β· sym β-zero-r   map : Hom (Coker.coapex (Ker.kernel f)) (Ker.ker (Coker.coeq f)) map = Ker.universal (Coker.coeq f) {e' = proj'}$ sym path


The existence of the map and indeed of the maps and follow from the universal properties of kernels and cokernels. The map is the canonical quotient map and the map is the canonical subobject inclusion

        where abstract
path : β.zeroβ β proj' β‘ Coker.coeq f β proj'
path = Coker.uniqueβ (Ker.kernel f)
{e' = 0m} (β-zero-r β sym β-zero-l)
(pushl (β.zero-βr _) β pulll ( apβ _β_ refl (β.hasβ€ _ .paths 0m)
β β-zero-r)
β β-zero-l)
(pullr (Coker.factors (Ker.kernel f)) β sym (Coker.coequal _)
β β-zero-r)

path =
Ker.kernel (Coker.coeq f) β map β Coker.coeq (Ker.kernel f) β‘β¨ pulll (Ker.factors _) β©β‘
proj' β Coker.coeq (Ker.kernel f)                           β‘β¨ Coker.factors _ β©β‘
f                                                           β


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

record is-abelian {o β} (C : Precategory o β) : Type (o β lsuc β) where
field has-is-preab : is-pre-abelian C
open is-pre-abelian has-is-preab public
field
coker-kerβker-coker
: β {A B} (f : Hom A B) β is-invertible (decompose f .fst)


This implies in particular that any monomorphism is a kernel, and every epimorphism is a cokernel. Letβs investigate the case for βevery mono is a kernelβ first: Suppose that is some monomorphism; Weβll show that itβs isomorphic to in the slice category

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


The map is obtained as the composite

where the isomorphism is our canonical map from before.

      fβkercoker : m.Hom (cut f) (cut (Ker.kernel (Coker.coeq f)))
fβkercoker ./-Hom.map = decompose f .fst β Coker.coeq (Ker.kernel f)
fβkercoker ./-Hom.commutes = sym (decompose f .snd)


Conversely, map is the composite

where the second map arises from the universal property of the cokernel: We can map out of it with the map since (using that is mono), we have from

      kercokerβf : m.Hom (cut (Ker.kernel (Coker.coeq f))) (cut f)
kercokerβf ./-Hom.map =
Coker.universal (Ker.kernel f) {e' = id} (monic _ _ path) β
coker-kerβker-coker f .is-invertible.inv
where abstract
path : f β id β 0m β‘ f β id β Ker.kernel f
path =
f β id β 0m              β‘β¨ ap (f β_) (eliml refl) β β-zero-r β©β‘
0m                       β‘Λβ¨ β.zero-βr _ β 0m-unique β©β‘Λ
(β.zeroβ β Ker.kernel f) β‘Λβ¨ Ker.equal f β©β‘Λ
f β Ker.kernel f         β‘β¨ ap (f β_) (introl refl) β©β‘
f β id β Ker.kernel f    β


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

      kercokerβf ./-Hom.commutes = path where
lemma =
is-coequaliserβis-epic (Coker.coeq _) (Coker.has-is-coeq _) _ _ $pullr (Coker.factors _) Β·Β· elimr refl Β·Β· (decompose f .snd β assoc _ _ _) path = invertibleβepic (coker-kerβker-coker _) _ _$
(f β Coker.universal _ _ β _) β decompose f .fst   β‘β¨ apβ _β_ (assoc _ _ _) refl β©β‘
((f β Coker.universal _ _) β _) β decompose f .fst β‘β¨ cancelr (coker-kerβker-coker _ .is-invertible.invr) β©β‘
f β Coker.universal _ _                            β‘β¨ lemma β©β‘
Ker.kernel _ β decompose f .fst                     β


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

    monoβkernel : cut f m.β cut (Ker.kernel (Coker.coeq f))
monoβkernel = m.make-iso fβkercoker kercokerβf fβkcβf kcβfβkc where
fβkcβf : fβkercoker m.β kercokerβf β‘ m.id
fβkcβf = ext $(decompose f .fst β Coker.coeq _) β Coker.universal _ _ β _ β‘β¨ cancel-inner lemma β©β‘ decompose f .fst β _ β‘β¨ coker-kerβker-coker f .is-invertible.invl β©β‘ id β where lemma = Coker.uniqueβ _ {e' = Coker.coeq (Ker.kernel f)} (β-zero-r β sym (sym (Coker.coequal _) β β-zero-r)) (pullr (Coker.factors (Ker.kernel f)) β elimr refl) (eliml refl) kcβfβkc : kercokerβf m.β fβkercoker β‘ m.id kcβfβkc = ext$
(Coker.universal _ _ β _) β decompose f .fst β Coker.coeq _ β‘β¨ cancel-inner (coker-kerβker-coker f .is-invertible.invr) β©β‘
Coker.universal _ _ β Coker.coeq _                          β‘β¨ Coker.factors _ β©β‘
id                                                          β