Bundles.agda

------------------------------------------------------------------------
-- The Agda standard library
--
-- Definitions of algebraic structures defined over some other
-- structure, like modules and vector spaces
--
-- Terminology of bundles:
-- * There are both *semimodules* and *modules*.
--   - For M an R-semimodule, R is a semiring, and M forms a commutative
--     monoid.
--   - For M an R-module, R is a ring, and M forms an Abelian group.
-- * There are all four of *left modules*, *right modules*, *bimodules*,
--   and *modules*.
--   - Left modules have a left-scaling operation.
--   - Right modules have a right-scaling operation.
--   - Bimodules have two sorts of scalars. Left-scaling handles one and
--     right-scaling handles the other. Left-scaling and right-scaling
--     are furthermore compatible.
--   - Modules are bimodules with a single sort of scalars and scalar
--     multiplication must also be commutative. Left-scaling and
--     right-scaling coincide.
------------------------------------------------------------------------

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

module Algebra.Module.Bundles where

open import Algebra.Bundles
open import Algebra.Core
open import Algebra.Definitions using (Involutive)
import Algebra.Module.Bundles.Raw as Raw
open import Algebra.Module.Core
open import Algebra.Module.Structures
open import Algebra.Module.Definitions
open import Algebra.Properties.Group
open import Function.Base
open import Level
open import Relation.Binary.Core using (Rel)
open import Relation.Nullary    using (¬_)
import Relation.Binary.Reasoning.Setoid as SetR

private
  variable
    r ℓr s ℓs : Level

------------------------------------------------------------------------
-- Re-export definitions of 'raw' bundles

open Raw public
  using ( RawLeftSemimodule; RawLeftModule
        ; RawRightSemimodule; RawRightModule
        ; RawBisemimodule; RawBimodule
        ; RawSemimodule; RawModule
        )

------------------------------------------------------------------------
-- Left modules
------------------------------------------------------------------------

record LeftSemimodule (semiring : Semiring r ℓr) m ℓm
                      : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
  open Semiring semiring

  infixr 7 _*ₗ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ₗ_ : Opₗ Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    isLeftSemimodule : IsLeftSemimodule semiring _≈ᴹ_ _+ᴹ_ 0ᴹ _*ₗ_

  open IsLeftSemimodule isLeftSemimodule public

  +ᴹ-commutativeMonoid : CommutativeMonoid m ℓm
  +ᴹ-commutativeMonoid = record
    { isCommutativeMonoid = +ᴹ-isCommutativeMonoid
    }

  open CommutativeMonoid +ᴹ-commutativeMonoid public
    using () renaming
    ( monoid    to +ᴹ-monoid
    ; semigroup to +ᴹ-semigroup
    ; magma     to +ᴹ-magma
    ; rawMagma  to +ᴹ-rawMagma
    ; rawMonoid to +ᴹ-rawMonoid
    ; _≉_ to _≉ᴹ_
    )

  rawLeftSemimodule : RawLeftSemimodule Carrier m ℓm
  rawLeftSemimodule = record
    { _≈ᴹ_ = _≈ᴹ_
    ; _+ᴹ_ = _+ᴹ_
    ; _*ₗ_ = _*ₗ_
    ; 0ᴹ = 0ᴹ
    }

record LeftModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
  open Ring ring

  infixr 8 -ᴹ_
  infixr 7 _*ₗ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ₗ_ : Opₗ Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    -ᴹ_ : Op₁ Carrierᴹ
    isLeftModule : IsLeftModule ring _≈ᴹ_ _+ᴹ_ 0ᴹ -ᴹ_ _*ₗ_

  open IsLeftModule isLeftModule public

  leftSemimodule : LeftSemimodule semiring m ℓm
  leftSemimodule = record { isLeftSemimodule = isLeftSemimodule }

  open LeftSemimodule leftSemimodule public
    using ( +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma
          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; rawLeftSemimodule; _≉ᴹ_)

  +ᴹ-abelianGroup : AbelianGroup m ℓm
  +ᴹ-abelianGroup = record { isAbelianGroup = +ᴹ-isAbelianGroup }

  open AbelianGroup +ᴹ-abelianGroup public
    using () renaming (group to +ᴹ-group; rawGroup to +ᴹ-rawGroup)

  rawLeftModule : RawLeftModule Carrier m ℓm
  rawLeftModule = record
    { _≈ᴹ_ = _≈ᴹ_
    ; _+ᴹ_ = _+ᴹ_
    ; _*ₗ_ = _*ₗ_
    ; 0ᴹ = 0ᴹ
    ; -ᴹ_ = -ᴹ_
    }

------------------------------------------------------------------------
-- Right modules
------------------------------------------------------------------------

record RightSemimodule (semiring : Semiring r ℓr) m ℓm
                       : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
  open Semiring semiring

  infixl 7 _*ᵣ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ᵣ_ : Opᵣ Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    isRightSemimodule : IsRightSemimodule semiring _≈ᴹ_ _+ᴹ_ 0ᴹ _*ᵣ_

  open IsRightSemimodule isRightSemimodule public

  +ᴹ-commutativeMonoid : CommutativeMonoid m ℓm
  +ᴹ-commutativeMonoid = record
    { isCommutativeMonoid = +ᴹ-isCommutativeMonoid
    }

  open CommutativeMonoid +ᴹ-commutativeMonoid public
    using () renaming
    ( monoid    to +ᴹ-monoid
    ; semigroup to +ᴹ-semigroup
    ; magma     to +ᴹ-magma
    ; rawMagma  to +ᴹ-rawMagma
    ; rawMonoid to +ᴹ-rawMonoid
    ; _≉_ to _≉ᴹ_
    )

  rawRightSemimodule : RawRightSemimodule Carrier m ℓm
  rawRightSemimodule = record
    { _≈ᴹ_ = _≈ᴹ_
    ; _+ᴹ_ = _+ᴹ_
    ; _*ᵣ_ = _*ᵣ_
    ; 0ᴹ = 0ᴹ
    }

record RightModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
  open Ring ring

  infixr 8 -ᴹ_
  infixl 7 _*ᵣ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ᵣ_ : Opᵣ Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    -ᴹ_ : Op₁ Carrierᴹ
    isRightModule : IsRightModule ring _≈ᴹ_ _+ᴹ_ 0ᴹ -ᴹ_ _*ᵣ_

  open IsRightModule isRightModule public

  rightSemimodule : RightSemimodule semiring m ℓm
  rightSemimodule = record { isRightSemimodule = isRightSemimodule }

  open RightSemimodule rightSemimodule public
    using ( +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma
          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; rawRightSemimodule; _≉ᴹ_)

  +ᴹ-abelianGroup : AbelianGroup m ℓm
  +ᴹ-abelianGroup = record { isAbelianGroup = +ᴹ-isAbelianGroup }

  open AbelianGroup +ᴹ-abelianGroup public
    using () renaming (group to +ᴹ-group; rawGroup to +ᴹ-rawGroup)

  rawRightModule : RawRightModule Carrier m ℓm
  rawRightModule = record
    { _≈ᴹ_ = _≈ᴹ_
    ; _+ᴹ_ = _+ᴹ_
    ; _*ᵣ_ = _*ᵣ_
    ; 0ᴹ = 0ᴹ
    ; -ᴹ_ = -ᴹ_
    }

------------------------------------------------------------------------
-- Bimodules
------------------------------------------------------------------------

record Bisemimodule (R-semiring : Semiring r ℓr) (S-semiring : Semiring s ℓs)
                    m ℓm : Set (r ⊔ s ⊔ ℓr ⊔ ℓs ⊔ suc (m ⊔ ℓm)) where
  private
    module R = Semiring R-semiring
    module S = Semiring S-semiring

  infixr 7 _*ₗ_
  infixl 7 _*ᵣ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ₗ_ : Opₗ R.Carrier Carrierᴹ
    _*ᵣ_ : Opᵣ S.Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    isBisemimodule : IsBisemimodule R-semiring S-semiring _≈ᴹ_ _+ᴹ_ 0ᴹ _*ₗ_ _*ᵣ_

  open IsBisemimodule isBisemimodule public

  leftSemimodule : LeftSemimodule R-semiring m ℓm
  leftSemimodule = record { isLeftSemimodule = isLeftSemimodule }

  rightSemimodule : RightSemimodule S-semiring m ℓm
  rightSemimodule = record { isRightSemimodule = isRightSemimodule }

  open LeftSemimodule leftSemimodule public
    using ( +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma; +ᴹ-rawMagma
          ; +ᴹ-rawMonoid; rawLeftSemimodule; _≉ᴹ_)

  open RightSemimodule rightSemimodule public
    using ( rawRightSemimodule )

  rawBisemimodule : RawBisemimodule R.Carrier S.Carrier m ℓm
  rawBisemimodule = record
    { _≈ᴹ_ = _≈ᴹ_
    ; _+ᴹ_ = _+ᴹ_
    ; _*ₗ_ = _*ₗ_
    ; _*ᵣ_ = _*ᵣ_
    ; 0ᴹ = 0ᴹ
    }

record Bimodule (R-ring : Ring r ℓr) (S-ring : Ring s ℓs) m ℓm
                : Set (r ⊔ s ⊔ ℓr ⊔ ℓs ⊔ suc (m ⊔ ℓm)) where
  private
    module R = Ring R-ring
    module S = Ring S-ring

  infixr 7 _*ₗ_
  infixl 7 _*ᵣ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ₗ_ : Opₗ R.Carrier Carrierᴹ
    _*ᵣ_ : Opᵣ S.Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    -ᴹ_ : Op₁ Carrierᴹ
    isBimodule : IsBimodule R-ring S-ring _≈ᴹ_ _+ᴹ_ 0ᴹ -ᴹ_ _*ₗ_ _*ᵣ_

  open IsBimodule isBimodule public

  leftModule : LeftModule R-ring m ℓm
  leftModule = record { isLeftModule = isLeftModule }

  rightModule : RightModule S-ring m ℓm
  rightModule = record { isRightModule = isRightModule }

  open LeftModule leftModule public
    using ( +ᴹ-abelianGroup; +ᴹ-commutativeMonoid; +ᴹ-group; +ᴹ-monoid
          ; +ᴹ-semigroup; +ᴹ-magma; +ᴹ-rawMagma; +ᴹ-rawMonoid; +ᴹ-rawGroup
          ; rawLeftSemimodule; rawLeftModule; _≉ᴹ_)

  open RightModule rightModule public
    using ( rawRightSemimodule; rawRightModule )

  bisemimodule : Bisemimodule R.semiring S.semiring m ℓm
  bisemimodule = record { isBisemimodule = isBisemimodule }

  open Bisemimodule bisemimodule public
    using (leftSemimodule; rightSemimodule; rawBisemimodule)

  rawBimodule : RawBimodule R.Carrier S.Carrier m ℓm
  rawBimodule = record
    { _≈ᴹ_ = _≈ᴹ_
    ; _+ᴹ_ = _+ᴹ_
    ; _*ₗ_ = _*ₗ_
    ; _*ᵣ_ = _*ᵣ_
    ; 0ᴹ = 0ᴹ
    ; -ᴹ_ = -ᴹ_
    }

------------------------------------------------------------------------
-- Modules over commutative structures
------------------------------------------------------------------------

record Semimodule (commutativeSemiring : CommutativeSemiring r ℓr) m ℓm
                  : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
  open CommutativeSemiring commutativeSemiring

  infixr 7 _*ₗ_
  infixl 7 _*ᵣ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ₗ_ : Opₗ Carrier Carrierᴹ
    _*ᵣ_ : Opᵣ Carrier Carrierᴹ
    0ᴹ : Carrierᴹ
    isSemimodule : IsSemimodule commutativeSemiring _≈ᴹ_ _+ᴹ_ 0ᴹ _*ₗ_ _*ᵣ_

  open IsSemimodule isSemimodule public

  private
    module L = LeftDefs Carrier _≈ᴹ_
    module R = RightDefs Carrier _≈ᴹ_

  bisemimodule : Bisemimodule semiring semiring m ℓm
  bisemimodule = record { isBisemimodule = isBisemimodule }

  open Bisemimodule bisemimodule public
    using ( leftSemimodule; rightSemimodule
          ; +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma
          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; rawLeftSemimodule; rawRightSemimodule
          ; rawBisemimodule; _≉ᴹ_
          )

  open SetR ≈ᴹ-setoid

  *ₗ-comm : L.Commutative _*ₗ_
  *ₗ-comm x y m = begin
    x *ₗ y *ₗ m   ≈⟨ ≈ᴹ-sym (*ₗ-assoc x y m) ⟩
    (x * y) *ₗ m  ≈⟨ *ₗ-cong (*-comm _ _) ≈ᴹ-refl ⟩
    (y * x) *ₗ m  ≈⟨ *ₗ-assoc y x m ⟩
    y *ₗ x *ₗ m   ∎

  *ᵣ-comm : R.Commutative _*ᵣ_
  *ᵣ-comm m x y = begin
    m *ᵣ x *ᵣ y   ≈⟨ *ᵣ-assoc m x y ⟩
    m *ᵣ (x * y)  ≈⟨ *ᵣ-cong ≈ᴹ-refl (*-comm _ _) ⟩
    m *ᵣ (y * x)  ≈⟨ ≈ᴹ-sym (*ᵣ-assoc m y x) ⟩
    m *ᵣ y *ᵣ x   ∎

  rawSemimodule : RawSemimodule Carrier m ℓm
  rawSemimodule = rawBisemimodule

record Module (commutativeRing : CommutativeRing r ℓr) m ℓm
              : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
  open CommutativeRing commutativeRing

  infixr 8 -ᴹ_
  infixr 7 _*ₗ_
  infixl 7 _*ᵣ_
  infixl 6 _+ᴹ_
  infix 4 _≈ᴹ_

  field
    Carrierᴹ : Set m
    _≈ᴹ_ : Rel Carrierᴹ ℓm
    _+ᴹ_ : Op₂ Carrierᴹ
    _*ₗ_ : Opₗ Carrier Carrierᴹ
    _*ᵣ_ : Opᵣ Carrier Carrierᴹ
    0ᴹ   : Carrierᴹ
    -ᴹ_  : Op₁ Carrierᴹ
    isModule : IsModule commutativeRing _≈ᴹ_ _+ᴹ_ 0ᴹ -ᴹ_ _*ₗ_ _*ᵣ_

  open IsModule isModule public

  bimodule : Bimodule ring ring m ℓm
  bimodule = record { isBimodule = isBimodule }

  open Bimodule bimodule public
    using ( leftModule; rightModule; leftSemimodule; rightSemimodule
          ; +ᴹ-abelianGroup; +ᴹ-group; +ᴹ-commutativeMonoid; +ᴹ-monoid
          ; +ᴹ-semigroup; +ᴹ-magma ; +ᴹ-rawMonoid; +ᴹ-rawMagma
          ; +ᴹ-rawGroup; rawLeftSemimodule; rawLeftModule; rawRightSemimodule
          ; rawRightModule; rawBisemimodule; rawBimodule; _≉ᴹ_)

  semimodule : Semimodule commutativeSemiring m ℓm
  semimodule = record { isSemimodule = isSemimodule }

  open Semimodule semimodule public using (*ₗ-comm; *ᵣ-comm; rawSemimodule)

  rawModule : RawModule Carrier m ℓm
  rawModule = rawBimodule