 Homomorphism Theorems

This is the Homomorphisms.Noether module of the Agda Universal Algebra Library.

{-# OPTIONS --without-K --exact-split --safe #-}

open import Algebras.Basic using ( 𝓞 ; 𝓥 ; Signature )

module Homomorphisms.Noether {𝑆 : Signature 𝓞 𝓥} where

-- Imports from Agda and the Agda Standard Library ---------------------------------------
open import Agda.Primitive  using ( Level ) renaming ( Set to Type )
open import Data.Product    using ( Σ-syntax ; _,_ )
renaming ( _×_ to _∧_ ; proj₁ to fst ; proj₂ to snd)
open import Function.Base   using ( _∘_ ; id )
open import Relation.Binary using ( IsEquivalence )
open import Relation.Binary.PropositionalEquality
using ( module ≡-Reasoning ; _≡_ ; cong ; refl ; cong-app )

-- Imports from agda-algebras --------------------------------------------------------------
open import Overture.Preliminaries        using ( ∣_∣ ; ∥_∥ ; _⁻¹ )
open import Overture.Inverses             using ( Image_∋_ )
open import Overture.Injective            using ( IsInjective )
open import Overture.Surjective           using ( IsSurjective ; SurjInv ; SurjInvIsInverseʳ )
open import Relations.Quotients           using ( ⌞_⌟ ; mkblk ; ⟪_⟫ )
open import Equality.Welldefined          using ( swelldef )
open import Equality.Truncation           using ( is-set ; blk-uip ; is-embedding ; monic-is-embedding|Set )
open import Equality.Extensionality       using ( pred-ext ; block-ext|uip )
open import Algebras.Basic                using ( Algebra ; _̂_)
open import Algebras.Congruences  {𝑆 = 𝑆} using ( Con ; IsCongruence )
open import Homomorphisms.Basic   {𝑆 = 𝑆} using ( hom ; is-homomorphism ; epi ; epi→hom )
open import Homomorphisms.Kernels {𝑆 = 𝑆} using ( kercon ; ker[_⇒_]_↾_ ; πker )
private variable α β γ : Level

The First Homomorphism Theorem

Here we formalize a version of the first homomorphism theorem, sometimes called Noether’s first homomorphism theorem, after Emmy Noether who was among the first proponents of the abstract approach to the subject that we now call “modern algebra”).

Informally, the theorem states that every homomorphism from 𝑨 to 𝑩 (𝑆-algebras) factors through the quotient algebra 𝑨 ╱ ker h (𝑨 modulo the kernel of the given homomorphism). In other terms, given h : hom 𝑨 𝑩 there exists φ : hom (𝑨 ╱ ker h) 𝑩 which, when composed with the canonical projection πker : 𝑨 ↠ 𝑨 ╱ ker h, is equal to h; that is, h = φ ∘ πker. Moreover, φ is a monomorphism (injective homomorphism) and is unique.

Our formal proof of this theorem will require function extensionality, proposition extensionality, and a couple of truncation assumptions. The extensionality assumptions are postulated using swelldef and pred-ext which were defined in Equality.Welldefined and Equality.Extensionality. As for truncation, to prove that φ is injective we require

• buip: uniqueness of (block) identity proofs; given two blocks of the kernel there is at most one proof that the blocks are equal;

To prove that φ is an embedding we require

• Bset: uniqueness of identity proofs in the codomain; that is, the codomain ∣ 𝑩 ∣ is assumed to be a set.

Note that the classical, informal statement of the first homomorphism theorem does not demand that φ be an embedding (in our sense of having subsingleton fibers), and if we left this out of the consequent of our formal theorem statement, then we could omit from the antecedent the assumption that ∣ 𝑩 ∣ is a set.

Without further ado, we present our formalization of the first homomorphism theorem.

open ≡-Reasoning

FirstHomTheorem|Set :

(𝑨 : Algebra α 𝑆)(𝑩 : Algebra β 𝑆)(h : hom 𝑨 𝑩)
(pe : pred-ext α β)(fe : swelldef 𝓥 β)                          -- extensionality assumptions
(Bset : is-set  𝑩 )(buip : blk-uip  𝑨   kercon fe {𝑩} h ) -- truncation assumptions
----------------------------------------------------------------
Σ[ φ  hom (ker[ 𝑨  𝑩 ] h  fe) 𝑩  ]
(  h    φ    πker fe{𝑩}h   IsInjective  φ     is-embedding  φ   )

FirstHomTheorem|Set 𝑨 𝑩 h pe fe Bset buip = (φ , φhom) , refl , φmon , φemb
where
θ : Con 𝑨
θ = kercon fe{𝑩} h
ξ : IsEquivalence  θ
ξ = IsCongruence.is-equivalence  θ

φ :  (ker[ 𝑨  𝑩 ] h  fe)    𝑩
φ a =  h   a

φhom : is-homomorphism (ker[ 𝑨  𝑩 ] h  fe) 𝑩 φ
φhom 𝑓 a =  h  ( (𝑓 ̂ 𝑨)  x   a x ) ) ≡⟨  h  𝑓  x   a x )
(𝑓 ̂ 𝑩) ( h    x   a x ))  ≡⟨ cong (𝑓 ̂ 𝑩) refl
(𝑓 ̂ 𝑩)  x  φ (a x))

φmon : IsInjective φ
φmon {_ , mkblk u refl} {_ , mkblk v refl} φuv = block-ext|uip pe buip ξ φuv

φemb : is-embedding φ
φemb = monic-is-embedding|Set φ Bset φmon

Below we will prove that the homomorphism φ, whose existence we just proved, is unique (see NoetherHomUnique), but first we show that if we add to the hypotheses of the first homomorphism theorem the assumption that h is surjective, then we obtain the so-called first isomorphism theorem. Naturally, we let FirstHomTheorem|Set do most of the work.

FirstIsoTheorem|Set :

(𝑨 : Algebra α 𝑆) (𝑩 : Algebra β 𝑆) (h : hom 𝑨 𝑩)
(pe : pred-ext α β) (fe : swelldef 𝓥 β)                        -- extensionality assumptions
(Bset : is-set  𝑩 )(buip : blk-uip  𝑨   kercon fe{𝑩}h )  -- truncation assumptions
IsSurjective  h
---------------------------------------------------------------
Σ[ f  (epi (ker[ 𝑨  𝑩 ] h  fe) 𝑩)]
(  h    f    πker fe{𝑩}h    IsInjective  f   is-embedding  f  )

FirstIsoTheorem|Set 𝑨 𝑩 h pe fe Bset buip hE =
(fmap , fhom , fepic) , refl , (snd  FHT )
where
FHT = FirstHomTheorem|Set 𝑨 𝑩 h pe fe Bset buip

fmap :  ker[ 𝑨  𝑩 ] h  fe    𝑩
fmap = fst  FHT

fhom : is-homomorphism (ker[ 𝑨  𝑩 ] h  fe) 𝑩 fmap
fhom = snd  FHT

fepic : IsSurjective fmap
fepic b = Goal where
a :  𝑨
a = SurjInv  h  hE b

bfa : b  fmap  a
bfa = ((SurjInvIsInverseʳ  h  hE) b)⁻¹

Goal : Image fmap  b
Goal = Image_∋_.eq  a  bfa

Now we prove that the homomorphism φ, whose existence is guaranteed by FirstHomTheorem|Set, is unique.

module _ {fe : swelldef 𝓥 β}(𝑨 : Algebra α 𝑆)(𝑩 : Algebra β 𝑆)(h : hom 𝑨 𝑩) where

FirstHomUnique : (f g : hom (ker[ 𝑨  𝑩 ] h  fe) 𝑩)
h    f    πker fe{𝑩}h
h    g    πker fe{𝑩}h
a     f  a   g  a

FirstHomUnique f g hfk hgk (_ , mkblk a refl) =
f  (_ , mkblk a refl) ≡⟨ cong-app(hfk ⁻¹)a
h  a                    ≡⟨ cong-app(hgk)a
g  (_ , mkblk a refl)

If, in addition, we postulate extensionality of functions defined on the domain ker[ 𝑨 ⇒ 𝑩 ] h, then we obtain the following variation of the last result. (See Equality.Truncation for a discussion of truncation, sets, and uniqueness of identity proofs.)

fe-FirstHomUnique : {fuww : funext (α ⊔ lsuc β) β}(f g : hom (ker[ 𝑨 ⇒ 𝑩 ] h ↾ fe) 𝑩)
→                    ∣ h ∣ ≡ ∣ f ∣ ∘ ∣ πker fe{𝑩}h ∣
→                    ∣ h ∣ ≡ ∣ g ∣ ∘ ∣ πker fe{𝑩}h ∣
→                    ∣ f ∣ ≡ ∣ g ∣

fe-FirstHomUnique {fuww} f g hfk hgk = fuww (NoetherHomUnique f g hfk hgk)

The proof of NoetherHomUnique goes through for the special case of epimorphisms, as we now verify.

FirstIsoUnique : (f g : epi (ker[ 𝑨  𝑩 ] h  fe) 𝑩)
h    f    πker fe{𝑩}h
h    g    πker fe{𝑩}h
a   f  a   g  a

FirstIsoUnique f g hfk hgk = FirstHomUnique (epi→hom 𝑩 f) (epi→hom 𝑩 g) hfk hgk