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 for whatever your favourite ring is1. The first thing we’ll show is that it admits an 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 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 with an structure, which certainly includes an abelian group structure, we can conclude that is not only a category, but an 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 is a preadditive category. This is exactly as in The zero object is the zero group, equipped with its unique choice of structure, and direct products are given by direct products of the underlying groups with the canonical choice of 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 structures of and
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∘factor = trivial! prod .has-is-product .π₂∘factor = trivial! prod .has-is-product .unique other p q = ext λ x → p #ₚ x , q #ₚ x