open import Algebra.Ring.Module.Notation
open import Algebra.Ring.Module.Action
open import Algebra.Ring.Commutative
open import Algebra.Group.Notation
open import Algebra.Ring.Module
open import Algebra.Group.Ab
open import Algebra.Group
open import Algebra.Ring

open import Cat.Displayed.Univalence.Thin
open import Cat.Abelian.Instances.Ab
open import Cat.Diagram.Terminal
open import Cat.Diagram.Product
open import Cat.Abelian.Base
open import Cat.Prelude hiding (_+_ ; _*_)

module Algebra.Ring.Module.Category {ℓ} (R : Ring ℓ) where

private module R = Ring-on (R .snd)
open Ab-category hiding (_+_)
open is-additive hiding (_+_)
open make-abelian-group
open Total-hom

open Module-notation ⦃ ... ⦄
open Additive-notation ⦃ ... ⦄

The category R-Mod🔗

Let us investigate the structure of the category $R\text{-Mod,}$ for whatever your favourite ring $R$ is¹. The first thing we’ll show is that it admits an $\mathbf{Ab}\text{-enrichment.}$ This is the usual “pointwise” group structure, but proving that the pointwise sum is a still a linear map is, ahem, very annoying. See for yourself:

module _ {ℓm ℓn} (M : Module R ℓm) (N : Module R ℓn) where
  instance
    _ = module-notation M
    _ = module-notation N

  +-is-linear-map
    : ∀ {f g : ⌞ M ⌟ → ⌞ N ⌟}
    → is-linear-map f (M .snd) (N .snd)
    → is-linear-map g (M .snd) (N .snd)
    → is-linear-map (λ x → f x + g x) (M .snd) (N .snd)
  +-is-linear-map {f = f} {g} fp gp .linear r s t =
    f (r ⋆ s + t) + g (r ⋆ s + t)      ≡⟨ ap₂ _+_ (fp .linear r s t) (gp .linear r s t) ⟩≡
    (r ⋆ f s + f t) + (r ⋆ g s + g t)  ≡⟨ sym +-assoc ∙ ap₂ _+_ refl (+-assoc ∙ ap₂ _+_ (+-comm _ _) refl ∙ sym +-assoc) ∙ +-assoc ∙ +-assoc ⟩≡
    ⌜ r ⋆ f s + r ⋆ g s ⌝ + f t + g t  ≡˘⟨ ap¡ (⋆-distribl r (f s) (g s)) ⟩≡˘
    r ⋆ (f s + g s) + f t + g t        ≡˘⟨ +-assoc ⟩≡˘
    r ⋆ (f s + g s) + (f t + g t)      ∎

Doing some further algebra will let us prove that linear maps are also closed under pointwise inverse, and contain the zero map. The calculations speak for themselves:

  neg-is-linear-map
    : ∀ {f : ⌞ M ⌟ → ⌞ N ⌟}
    → is-linear-map f (M .snd) (N .snd)
    → is-linear-map (λ x → - f x) (M .snd) (N .snd)
  neg-is-linear-map {f = f} fp .linear r s t =
    - f (r ⋆ s + t)        ≡⟨ ap -_ (fp .linear r s t) ⟩≡
    - (r ⋆ f s + f t)      ≡⟨ neg-comm ∙ +-comm _ _ ⟩≡
    - (r ⋆ f s) + - (f t)  ≡⟨ ap₂ _+_ (sym (Module-on.⋆-invr (N .snd))) refl ⟩≡
    r ⋆ - f s + - f t      ∎

  0-is-linear-map : is-linear-map (λ x → 0g) (M .snd) (N .snd)
  0-is-linear-map .linear r s t = sym (+-idr ∙ Module-on.⋆-idr (N .snd))

Finally, if the base ring $R$ is commutative, then linear maps are also closed under pointwise scalar multiplication:

  ⋆-is-linear-map
    : ∀ {f : ⌞ M ⌟ → ⌞ N ⌟} {r : ⌞ R ⌟}
    → is-commutative-ring R
    → is-linear-map f (M .snd) (N .snd)
    → is-linear-map (λ x → r ⋆ f x) (M .snd) (N .snd)
  ⋆-is-linear-map {f = f} {r} cring fp .linear s x y =
    r ⋆ f (s ⋆ x + y)      ≡⟨ ap (r ⋆_) (fp .linear _ _ _) ⟩≡
    r ⋆ (s ⋆ f x + f y)    ≡⟨ ⋆-distribl r (s ⋆ f x) (f y) ⟩≡
    r ⋆ s ⋆ f x + r ⋆ f y  ≡⟨ ap (_+ r ⋆ f y) (⋆-assoc _ _ _ ∙ ap (_⋆ f x) cring ∙ sym (⋆-assoc _ _ _)) ⟩≡
    s ⋆ r ⋆ f x + r ⋆ f y  ∎

  Linear-map-group : Abelian-group (ℓ ⊔ ℓm ⊔ ℓn)
  ∣ Linear-map-group .fst ∣ = Linear-map M N
  Linear-map-group .fst .is-tr = Linear-map-is-set R
  Linear-map-group .snd = to-abelian-group-on grp where
    grp : make-abelian-group (Linear-map M N)
    grp .ab-is-set = Linear-map-is-set R

    grp .mul f g .map x = f .map x + g .map x
    grp .mul f g .lin = +-is-linear-map (f .lin) (g .lin)

    grp .inv f .map x = - f .map x
    grp .inv f .lin = neg-is-linear-map (f .lin)

    grp .1g .map x = 0g
    grp .1g .lin = 0-is-linear-map

    grp .idl f       = ext λ x → +-idl
    grp .assoc f g h = ext λ x → +-assoc
    grp .invl f      = ext λ x → +-invl
    grp .comm f g    = ext λ x → +-comm _ _

module _ (cring : is-commutative-ring R) {ℓm ℓn} (M : Module R ℓm) (N : Module R ℓn) where
  private instance
    _ = module-notation M
    _ = module-notation N

  Action-on-hom : Ring-action R (Linear-map-group M N .snd)
  Action-on-hom .Ring-action._⋆_ r f .map z = r ⋆ f .map z
  Action-on-hom .Ring-action._⋆_ r f .lin = ⋆-is-linear-map M N cring (f .lin)
  Action-on-hom .Ring-action.⋆-distribl f g h = ext λ x → ⋆-distribl _ _ _
  Action-on-hom .Ring-action.⋆-distribr f g h = ext λ x → ⋆-distribr _ _ _
  Action-on-hom .Ring-action.⋆-assoc f g h = ext λ x → ⋆-assoc _ _ _
  Action-on-hom .Ring-action.⋆-id f = ext λ x → ⋆-id _

  Hom-Mod : Module R (level-of ⌞ R ⌟ ⊔ ℓm ⊔ ℓn)
  Hom-Mod .fst = Action→Module R (Linear-map-group M N) Action-on-hom .fst
  Hom-Mod .snd = Action→Module R (Linear-map-group M N) Action-on-hom .snd

Since we’ve essentially equipped the set of linear maps $M\to N$ with an $R\text{-module}$ structure, which certainly includes an abelian group structure, we can conclude that $R\text{-}\mathbf{Mod}$ is not only a category, but an $\mathbf{Ab}\text{-category}$ to boot!

R-Mod-ab-category : ∀ {ℓ'} → Ab-category (R-Mod R ℓ')

R-Mod-ab-category .Abelian-group-on-hom A B = to-abelian-group-on grp where
  instance
    _ = module-notation A
    _ = module-notation B

  grp : make-abelian-group (R-Mod.Hom A B)
  grp .ab-is-set = R-Mod.Hom-set _ _
  grp .mul f g .hom x = f # x + g # x
  grp .mul f g .preserves = +-is-linear-map A B (f .preserves) (g .preserves)
  grp .inv f .hom x = - f # x
  grp .inv f .preserves = neg-is-linear-map A B (f .preserves)
  grp .1g .hom x = 0g
  grp .1g .preserves = 0-is-linear-map A B

  grp .idl f       = ext λ x → +-idl
  grp .assoc f g h = ext λ x → +-assoc
  grp .invl f      = ext λ x → +-invl
  grp .comm f g    = ext λ x → +-comm _ _

R-Mod-ab-category .∘-linear-l f g h = trivial!
R-Mod-ab-category .∘-linear-r {B = B} {C} f g h = ext λ x →
  sym (is-linear-map.pres-+ (f .preserves) _ _)

Finite biproducts🔗

Let’s now prove that $R\text{-Mod}$ is a preadditive category. This is exactly as in $\mathbf{Ab}\text{:}$ The zero object is the zero group, equipped with its unique choice of $R\text{-module}$ structure, and direct products $M\oplus N$ are given by direct products of the underlying groups $M_G\oplus N_G$ with the canonical choice of $R\text{-module}$ structure.

The zero object is simple, because the unit type is so well-behaved² when it comes to definitional equality: Everything is constantly the unit, including the paths, which are all reflexivity.

R-Mod-is-additive : ∀ {ℓ} → is-additive (R-Mod R ℓ)
R-Mod-is-additive .has-ab = R-Mod-ab-category
R-Mod-is-additive .has-terminal = term where
  act : Ring-action R _
  act .Ring-action._⋆_ r _          = lift tt
  act .Ring-action.⋆-distribl r x y = refl
  act .Ring-action.⋆-distribr r x y = refl
  act .Ring-action.⋆-assoc r s x    = refl
  act .Ring-action.⋆-id x           = refl

  ∅ᴹ : Module R _
  ∅ᴹ = Action→Module R (Ab-is-additive .has-terminal .Terminal.top) act

  term : Terminal (R-Mod R _)
  term .Terminal.top = ∅ᴹ
  term .Terminal.has⊤ x .centre .hom _ = lift tt
  term .Terminal.has⊤ x .centre .preserves .linear r s t = refl
  term .Terminal.has⊤ x .paths r = trivial!

For the direct products, on the other hand, we have to do a bit more work. Like we mentioned before, the direct product of modules is built on the direct product of abelian groups (which is, in turn, built on the Cartesian product of types). The module action, and its laws, are defined pointwise using the $R\text{-module}$ structures of $M$ and $N\text{:}$

R-Mod-is-additive .has-prods M N = prod where
  module P = Product (Ab-is-additive .has-prods
    (M .fst , Module-on→Abelian-group-on (M .snd))
    (N .fst , Module-on→Abelian-group-on (N .snd)))

  instance
    _ = module-notation M
    _ = module-notation N

  act : Ring-action R _
  act .Ring-action._⋆_ r (a , b)    = r ⋆ a , r ⋆ b
  act .Ring-action.⋆-distribl r x y = ap₂ _,_ (⋆-distribl _ _ _) (⋆-distribl _ _ _)
  act .Ring-action.⋆-distribr r x y = ap₂ _,_ (⋆-distribr _ _ _) (⋆-distribr _ _ _)
  act .Ring-action.⋆-assoc r s x    = ap₂ _,_ (⋆-assoc _ _ _) (⋆-assoc _ _ _)
  act .Ring-action.⋆-id x           = ap₂ _,_ (⋆-id _) (⋆-id _)

  M⊕ᵣN : Module R _
  M⊕ᵣN = Action→Module R P.apex act

We can readily define the universal cone: The projection maps are the projection maps of the underlying type, which are definitionally linear. Proving that this cone is actually universal involves a bit of path-mangling, but it’s nothing too bad:

  open Product
  open is-product

  prod : Product (R-Mod _ _) M N
  prod .apex = M⊕ᵣN
  prod .π₁ .hom = fst
  prod .π₁ .preserves .linear r s t = refl
  prod .π₂ .hom = snd
  prod .π₂ .preserves .linear r s t = refl
  prod .has-is-product .⟨_,_⟩ f g .hom x = f # x , g # x
  prod .has-is-product .⟨_,_⟩ f g .preserves .linear r m s =
    Σ-pathp (f .preserves .linear _ _ _) (g .preserves .linear _ _ _)
  prod .has-is-product .π₁∘⟨⟩ = trivial!
  prod .has-is-product .π₂∘⟨⟩ = trivial!
  prod .has-is-product .unique p q = ext λ x → p #ₚ x ,ₚ q #ₚ x