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Legacy.Base.Structures.Sigma.Homs

Homomorphisms of general structures

{-# OPTIONS --cubical-compatible --exact-split --safe #-}

module Legacy.Base.Structures.Sigma.Homs where

open import Agda.Primitive  using () renaming ( Set to Type )

-- Imports from the Agda Standard Library ----------------------------------------------------------
open import Data.Product    using ( _,_ ; _Γ—_ ; Ξ£-syntax ) renaming ( proj₁ to fst ; projβ‚‚ to snd )
open import Level           using ( _βŠ”_ ; Level ; Lift ; lift ; lower ) renaming ( 0β„“ to β„“β‚€ ; suc to lsuc )
open import Function.Base   using ( _∘_ ; id )
open import Relation.Binary.PropositionalEquality
                            using ( _≑_ ;  cong ; refl ; module ≑-Reasoning )

-- Imports from the Agda Universal Algebra Library ---------------------------------------------
open import Overture        using ( ∣_∣ ; βˆ₯_βˆ₯ ; _βˆ™_ ; _⁻¹)
open import Legacy.Base.Functions  using ( IsInjective ; IsSurjective )
open import Legacy.Base.Relations  using ( _|:_ ; 0[_] ; ker ; Equivalence ; Quotient )
                            using ( 0[_]Equivalence ; ker-IsEquivalence ; βŸͺ_⟫ )
                            using ( kerlift-IsEquivalence ; ⌞_⌟ ; βŸͺ_∼_⟫-elim ; _/_ )
open import Legacy.Base.Equality   using ( swelldef )
open import Legacy.Base.Structures.Sigma.Basic
                            using ( Signature ; Structure ; Compatible ; _Κ³_ ; _α΅’_ )
                            using ( Lift-StrucΚ³ ; Lift-StrucΛ‘ ; Lift-Struc )

private variable 𝑅 𝐹 : Signature

-- Development for Structures (Sigma type representation)
module _  {Ξ± ρᡃ : Level} (𝑨 : Structure  𝑅 𝐹 {Ξ±}{ρᡃ})
          {Ξ² ρᡇ : Level} (𝑩 : Structure 𝑅 𝐹 {Ξ²}{ρᡇ}) where

 preserves : ∣ 𝑅 ∣ β†’ (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” ρᡃ βŠ” ρᡇ)
 preserves r h = βˆ€ a β†’ ((r Κ³ 𝑨) a) β†’ ((r Κ³ 𝑩) (h ∘ a))

 is-hom-rel : (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” ρᡃ βŠ” ρᡇ)
 is-hom-rel h = βˆ€ r β†’  preserves r h

 comp-op : ∣ 𝐹 ∣ β†’ (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” Ξ²)
 comp-op f h = βˆ€ a β†’ h ((f α΅’ 𝑨) a) ≑ (f α΅’ 𝑩) (h ∘ a)

 is-hom-op : (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” Ξ²)
 is-hom-op h = βˆ€ f β†’ comp-op f h

 is-hom : (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” ρᡃ βŠ” Ξ² βŠ” ρᡇ)
 is-hom h = is-hom-rel h Γ— is-hom-op h

 hom : Type (Ξ± βŠ” ρᡃ βŠ” Ξ² βŠ” ρᡇ)
 hom = Ξ£[ h ∈ (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) ] is-hom h

module _  {𝑅 𝐹 : Signature}
          {Ξ± ρᡃ : Level}(𝑨 : Structure 𝑅 𝐹 {Ξ±}{ρᡃ})
          {Ξ² ρᡇ : Level}{𝑩 : Structure 𝑅 𝐹 {Ξ²}{ρᡇ}}
          {Ξ³ ρᢜ : Level}(π‘ͺ : Structure 𝑅 𝐹 {Ξ³}{ρᢜ}) where

 ∘-is-hom-rel :  {f : ∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣}{g : ∣ 𝑩 ∣ β†’ ∣ π‘ͺ ∣}
  β†’              is-hom-rel 𝑨 𝑩 f β†’ is-hom-rel 𝑩 π‘ͺ g β†’ is-hom-rel 𝑨 π‘ͺ (g ∘ f)

 ∘-is-hom-rel {f}{g} fhr ghr R a = Ξ» z β†’ ghr R (Ξ» z₁ β†’ f (a z₁)) (fhr R a z)


 ∘-is-hom-op :  {f : ∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣}{g : ∣ 𝑩 ∣ β†’ ∣ π‘ͺ ∣}
  β†’             is-hom-op 𝑨 𝑩 f β†’ is-hom-op 𝑩 π‘ͺ g β†’ is-hom-op 𝑨 π‘ͺ (g ∘ f)

 ∘-is-hom-op {f}{g} fho gho 𝑓 a = cong g (fho 𝑓 a) βˆ™ gho 𝑓 (f ∘ a)


 ∘-is-hom :  {f : ∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣}{g : ∣ 𝑩 ∣ β†’ ∣ π‘ͺ ∣}
  β†’          is-hom 𝑨 𝑩 f β†’ is-hom 𝑩 π‘ͺ g β†’ is-hom 𝑨 π‘ͺ (g ∘ f)

 ∘-is-hom {f} {g} fhro ghro = ihr , iho
  where
  ihr : is-hom-rel 𝑨 π‘ͺ (g ∘ f)
  ihr = ∘-is-hom-rel {f}{g} (fst fhro) (fst ghro)

  iho : is-hom-op 𝑨 π‘ͺ (g ∘ f)
  iho = ∘-is-hom-op {f}{g} (snd fhro) (snd ghro)

 ∘-hom : hom 𝑨 𝑩 β†’ hom 𝑩 π‘ͺ β†’ hom 𝑨 π‘ͺ
 ∘-hom (f , fh) (g , gh) = g ∘ f , ∘-is-hom {f}{g} fh gh


module _ {α ρ : Level} where

 𝒾𝒹 : (𝑨 : Structure 𝑅 𝐹 {Ξ±}{ρ}) β†’ hom 𝑨 𝑨
 𝒾𝒹 _ = id , (Ξ» R a z β†’ z)  , (Ξ» f a β†’ refl)

module _  {Ξ± ρᡃ : Level} (𝑨 : Structure 𝑅 𝐹 {Ξ±}{ρᡃ})
          {Ξ² ρᡇ : Level} (𝑩 : Structure 𝑅 𝐹 {Ξ²}{ρᡇ}) where

 is-mon : (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” ρᡃ βŠ” Ξ² βŠ” ρᡇ)
 is-mon g = is-hom 𝑨 𝑩 g Γ— IsInjective g

 mon : Type (Ξ± βŠ” ρᡃ βŠ” Ξ² βŠ” ρᡇ)
 mon = Ξ£[ g ∈ (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) ] is-mon g

 is-epi : (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) β†’ Type (Ξ± βŠ” ρᡃ βŠ” Ξ² βŠ” ρᡇ)
 is-epi g = is-hom 𝑨 𝑩 g Γ— IsSurjective g

 epi : Type (Ξ± βŠ” ρᡃ βŠ” Ξ² βŠ” ρᡇ)
 epi = Ξ£[ g ∈ (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) ] is-epi g

 monβ†’hom : mon β†’ hom 𝑨 𝑩
 mon→hom ϕ = (fst ϕ) , fst (snd ϕ )

 epiβ†’hom : epi β†’ hom 𝑨 𝑩
 epi→hom ϕ = (fst ϕ) , fst (snd ϕ)

Next, lift and lower are (the maps of) homomorphisms.

module _ {𝑅 𝐹 : Signature}{Ξ± ρᡃ : Level} where
 open Lift

 𝓁𝒾𝒻𝓉 : (β„“ ρ : Level)(𝑨 : Structure  𝑅 𝐹{Ξ±}{ρᡃ}) β†’ hom 𝑨 (Lift-Struc β„“ ρ 𝑨)
 𝓁𝒾𝒻𝓉 = Ξ» β„“ ρ 𝑨 β†’ lift , ( (Ξ» R a x β†’ lift x) , Ξ» f a β†’ refl )

 π“β„΄π“Œβ„―π“‡ : (β„“ ρ : Level)(𝑨 : Structure  𝑅 𝐹{Ξ±}{ρᡃ}) β†’ hom (Lift-Struc β„“ ρ 𝑨) 𝑨
 π“β„΄π“Œβ„―π“‡ = Ξ» β„“ ρ 𝑨 β†’ lower , (Ξ» R a x β†’ lower x) , (Ξ» f a β†’ refl)

module _  {𝑅 𝐹 : Signature}{Ξ± ρᡃ Ξ² ρᡇ : Level}{𝑅 𝐹 : Signature}
          {𝑨 : Structure 𝑅 𝐹 {Ξ±}{ρᡃ}}{𝑩 : Structure 𝑅 𝐹 {Ξ²}{ρᡇ}} where

 Lift-Hom : (β„“ ρ β„“' ρ' : Level) β†’ hom 𝑨 𝑩 β†’ hom (Lift-Struc β„“ ρ 𝑨) (Lift-Struc β„“' ρ' 𝑩)
 Lift-Hom β„“ ρ β„“' ρ' (h , hhom) = lift ∘ h ∘ lower , Goal
  where
  lABh : is-hom (Lift-Struc β„“ ρ 𝑨) 𝑩 (h ∘ lower)
  lABh = ∘-is-hom{𝑅 = 𝑅}{𝐹} (Lift-Struc β„“ ρ 𝑨) 𝑩{lower}{h} ((Ξ» R a x β†’ lower x) , (Ξ» f a β†’ refl)) hhom

  Goal : is-hom (Lift-Struc β„“ ρ 𝑨) (Lift-Struc β„“' ρ' 𝑩) (lift ∘ h ∘ lower)
  Goal = ∘-is-hom  {𝑅 = 𝑅}{𝐹} (Lift-Struc β„“ ρ 𝑨) (Lift-Struc β„“' ρ' 𝑩)
                   {h ∘ lower}{lift} lABh ((Ξ» R a x β†’ lift x) , (Ξ» f a β†’ refl))

Kernels of homomorphisms of structures of sigma type

The kernel of a homomorphism is a congruence relation and conversely for every congruence relation ΞΈ, there exists a homomorphism with kernel ΞΈ (namely, that canonical projection onto the quotient modulo ΞΈ).

open ≑-Reasoning
module _  {𝑅 𝐹 : Signature} {Ξ± ρᡃ Ξ² ρᡇ : Level}
          {𝑨 : Structure 𝑅 𝐹 {Ξ±}{ρᡃ}}{𝑩 : Structure 𝑅 𝐹{Ξ²}{ρᡇ}} where

 Homker-comp : swelldef β„“β‚€ Ξ² β†’ (h : hom 𝑨 𝑩) β†’ Compatible 𝑨 (ker ∣ h ∣)
 Homker-comp wd h f {u}{v} kuv =  (∣ h ∣ ((f α΅’ 𝑨) u))   β‰‘βŸ¨(snd βˆ₯ h βˆ₯) f u ⟩
                                  ((f α΅’ 𝑩)(∣ h ∣ ∘ u))  β‰‘βŸ¨ wd (f α΅’ 𝑩) (∣ h ∣ ∘ u) (∣ h ∣ ∘ v) kuv ⟩
                                  ((f α΅’ 𝑩)(∣ h ∣ ∘ v))  β‰‘βŸ¨((snd βˆ₯ h βˆ₯) f v)⁻¹ ⟩
                                  (∣ h ∣((f α΅’ 𝑨) v))    ∎


(Notice, it is here that the swelldef postulate comes into play, and because it is needed to prove homker-comp, it is postulated by all the lemmas below that depend upon homker-comp.)

It is convenient to define a function that takes a homomorphism and constructs a congruence from its kernel. We call this function kercon.

kercon : swelldef π“₯ 𝓦 β†’ {𝑩 : Algebra 𝓦 𝑆} β†’ hom 𝑨 𝑩 β†’ Con{𝓀}{𝓦} 𝑨 kercon wd {𝑩} h = ker ∣ h ∣ , mkcon (ker-IsEquivalence ∣ h ∣)(homker-comp wd {𝑩} h)

\end{code}

With this congruence we construct the corresponding quotient, along with some syntactic sugar to denote it.

kerquo : swelldef π“₯ 𝓦 β†’ {𝑩 : Algebra 𝓦 𝑆} β†’ hom 𝑨 𝑩 β†’ Algebra (𝓀 βŠ” lsuc 𝓦) 𝑆 kerquo wd {𝑩} h = 𝑨 β•± (kercon wd {𝑩} h)

ker[β‡’]β†Ύ : (𝑨 : Algebra 𝓀 𝑆)(𝑩 : Algebra 𝓦 𝑆) β†’ hom 𝑨 𝑩 β†’ swelldef π“₯ 𝓦 β†’ Algebra (𝓀 βŠ” lsuc 𝓦) 𝑆 ker[ 𝑨 β‡’ 𝑩 ] h β†Ύ wd = kerquo wd {𝑩} h

\end{code}

Thus, given h : hom 𝑨 𝑩, we can construct the quotient of 𝑨 modulo the kernel of h, and the syntax for this quotient in the agda-algebras library is 𝑨 [ 𝑩 ]/ker h β†Ύ fe.

The canonical projection

Given an algebra 𝑨 and a congruence ΞΈ, the canonical projection is a map from 𝑨 onto 𝑨 β•± ΞΈ that is constructed, and proved epimorphic, as follows.

module _ {𝓀 𝓦 : Level}{𝑨 : Algebra 𝓀 𝑆} where Ο€epi : (ΞΈ : Con{𝓀}{𝓦} 𝑨) β†’ epi 𝑨 (𝑨 β•± ΞΈ) Ο€epi ΞΈ = (Ξ» a β†’ βŸͺ a ⟫) , (Ξ» _ _ β†’ refl) , cΟ€-is-epic where cΟ€-is-epic : IsSurjective (Ξ» a β†’ βŸͺ a ⟫) cΟ€-is-epic (C , (a , refl)) = Image_βˆ‹_.im a

\end{code}

In may happen that we don't care about the surjectivity of Ο€epi, in which case would might prefer to work with the homomorphic reduct of Ο€epi. This is obtained by applying epi-to-hom, like so.

Ο€hom : (ΞΈ : Con{𝓀}{𝓦} 𝑨) β†’ hom 𝑨 (𝑨 β•± ΞΈ) Ο€hom ΞΈ = epi-to-hom (𝑨 β•± ΞΈ) (Ο€epi ΞΈ)

\end{code}

We combine the foregoing to define a function that takes 𝑆-algebras 𝑨 and 𝑩, and a homomorphism h : hom 𝑨 𝑩 and returns the canonical epimorphism from 𝑨 onto 𝑨 [ 𝑩 ]/ker h. (Recall, the latter is the special notation we defined above for the quotient of 𝑨 modulo the kernel of h.)

Ο€ker : (wd : swelldef π“₯ 𝓦){𝑩 : Algebra 𝓦 𝑆}(h : hom 𝑨 𝑩) β†’ epi 𝑨 (ker[ 𝑨 β‡’ 𝑩 ] h β†Ύ wd) Ο€ker wd {𝑩} h = Ο€epi (kercon wd {𝑩} h)

\end{code}

The kernel of the canonical projection of 𝑨 onto 𝑨 / ΞΈ is equal to ΞΈ, but since equality of inhabitants of certain types (like Congruence or Rel) can be a tricky business, we settle for proving the containment 𝑨 / ΞΈ βŠ† ΞΈ. Of the two containments, this is the easier one to prove; luckily it is also the one we need later.

open IsCongruence

ker-in-con : {wd : swelldef π“₯ (𝓀 βŠ” lsuc 𝓦)}(ΞΈ : Con 𝑨) β†’ βˆ€ {x}{y} β†’ ∣ kercon wd {𝑨 β•± ΞΈ} (Ο€hom ΞΈ) ∣ x y β†’ ∣ ΞΈ ∣ x y

ker-in-con ΞΈ hyp = /-≑ ΞΈ hyp

\end{code}

Product homomorphisms

Suppose we have an algebra 𝑨, a type I : Type π“˜, and a family ℬ : I β†’ Algebra 𝓦 𝑆 of algebras. We sometimes refer to the inhabitants of I as indices, and call ℬ an indexed family of algebras.

If in addition we have a family 𝒽 : (i : I) β†’ hom 𝑨 (ℬ i) of homomorphisms, then we can construct a homomorphism from 𝑨 to the product β¨… ℬ in the natural way.

module _ {π“˜ 𝓦 : Level}{I : Type π“˜}(ℬ : I β†’ Algebra 𝓦 𝑆) where

β¨…-hom-co : funext π“˜ 𝓦 β†’ {𝓀 : Level}(𝑨 : Algebra 𝓀 𝑆) β†’ (βˆ€(i : I) β†’ hom 𝑨 (ℬ i)) β†’ hom 𝑨 (β¨… ℬ) β¨…-hom-co fe 𝑨 𝒽 = (Ξ» a i β†’ ∣ 𝒽 i ∣ a) , (Ξ» 𝑓 𝒢 β†’ fe Ξ» i β†’ βˆ₯ 𝒽 i βˆ₯ 𝑓 𝒢)

\end{code}

The family 𝒽 of homomorphisms inhabits the dependent type Ξ  i κž‰ I , hom 𝑨 (ℬ i). The syntax we use to represent this type is available to us because of the way -Ξ  is defined in the Type Topology library. We like this syntax because it is very close to the notation one finds in the standard type theory literature. However, we could equally well have used one of the following alternatives, which may be closer to "standard Agda" syntax:

Ξ  Ξ» i β†’ hom 𝑨 (ℬ i)   or   (i : I) β†’ hom 𝑨 (ℬ i)   or   βˆ€ i β†’ hom 𝑨 (ℬ i).

The foregoing generalizes easily to the case in which the domain is also a product of a family of algebras. That is, if we are given π’œ : I β†’ Algebra 𝓀 𝑆 and ℬ : I β†’ Algebra 𝓦 𝑆 (two families of 𝑆-algebras), and 𝒽 : Ξ  i κž‰ I , hom (π’œ i)(ℬ i) (a family of homomorphisms), then we can construct a homomorphism from β¨… π’œ to β¨… ℬ in the following natural way.

β¨…-hom : funext π“˜ 𝓦 β†’ {𝓀 : Level}(π’œ : I β†’ Algebra 𝓀 𝑆) β†’ Ξ [ i κž‰ I ] hom (π’œ i)(ℬ i) β†’ hom (β¨… π’œ)(β¨… ℬ) β¨…-hom fe π’œ 𝒽 = (Ξ» x i β†’ ∣ 𝒽 i ∣ (x i)) , (Ξ» 𝑓 𝒢 β†’ fe Ξ» i β†’ βˆ₯ 𝒽 i βˆ₯ 𝑓 (Ξ» x β†’ 𝒢 x i))

\end{code}

Projection out of products

Later we will need a proof of the fact that projecting out of a product algebra onto one of its factors is a homomorphism.

β¨…-projection-hom : Ξ [ i κž‰ I ] hom (β¨… ℬ) (ℬ i) β¨…-projection-hom = Ξ» x β†’ (Ξ» z β†’ z x) , Ξ» _ _ β†’ refl

\end{code}

We could prove a more general result involving projections onto multiple factors, but so far the single-factor result has sufficed.