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reorganising category_theory/instances/rings.lean
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‎src/category_theory/instances/CommRing/adjunctions.lean

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/- Copyright (c) 2019 Scott Morrison. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Scott Morrison, Johannes Hölzl
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Multivariable polynomials on a type is the left adjoint of the
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forgetful functor from commutative rings to types.
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-/
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import category_theory.instances.CommRing.basic
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import category_theory.adjunction
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import data.mv_polynomial
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universe u
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open mv_polynomial
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open category_theory
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open category_theory.instances
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namespace category_theory.instances.CommRing
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local attribute [instance, priority 0] subtype.fintype set_fintype classical.prop_decidable
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noncomputable def polynomial_ring : Type u ⥤ CommRing.{u} :=
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{ obj := λ α, ⟨mv_polynomial α ℤ, by apply_instance⟩,
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map := λ α β f, ⟨eval₂ C (X ∘ f), by apply_instance⟩,
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map_id' := λ α, subtype.ext.mpr $ funext $ eval₂_eta,
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map_comp' := λ α β γ f g, subtype.ext.mpr $ funext $ λ p,
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by apply mv_polynomial.induction_on p; intros;
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simp only [*, eval₂_add, eval₂_mul, eval₂_C, eval₂_X, comp_val,
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eq_self_iff_true, function.comp_app, types_comp] at * }
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@[simp] lemma polynomial_ring_obj_α {α : Type u} :
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(polynomial_ring.obj α).α = mv_polynomial α ℤ := rfl
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@[simp] lemma polynomial_ring_map_val {α β : Type u} {f : α → β} :
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(polynomial_ring.map f).val = eval₂ C (X ∘ f) := rfl
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noncomputable def adj : adjunction polynomial_ring (forget : CommRing ⥤ Type u) :=
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adjunction.mk_of_hom_equiv _ _
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{ hom_equiv := λ α R,
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{ to_fun := λ f, f ∘ X,
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inv_fun := λ f, ⟨eval₂ int.cast f, by apply_instance⟩,
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left_inv := λ f, subtype.ext.mpr $ funext $ λ p,
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begin
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have H0 := λ n, (congr (int.eq_cast' (f.val ∘ C)) (rfl : n = n)).symm,
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have H1 := λ p₁ p₂, (@is_ring_hom.map_add _ _ _ _ f.val f.2 p₁ p₂).symm,
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have H2 := λ p₁ p₂, (@is_ring_hom.map_mul _ _ _ _ f.val f.2 p₁ p₂).symm,
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apply mv_polynomial.induction_on p; intros;
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simp only [*, eval₂_add, eval₂_mul, eval₂_C, eval₂_X,
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eq_self_iff_true, function.comp_app, hom_coe_app] at *
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end,
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right_inv := by tidy },
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hom_equiv_naturality_left_symm' := λ X' X Y f g, subtype.ext.mpr $ funext $ λ p,
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begin
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apply mv_polynomial.induction_on p; intros;
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simp only [*, eval₂_mul, eval₂_add, eval₂_C, eval₂_X,
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comp_val, equiv.coe_fn_symm_mk, hom_coe_app, polynomial_ring_map_val,
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eq_self_iff_true, function.comp_app, add_right_inj, types_comp] at *
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end }
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end category_theory.instances.CommRing

‎src/category_theory/instances/rings.lean renamed to ‎src/category_theory/instances/CommRing/basic.lean

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@@ -3,15 +3,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Scott Morrison, Johannes Hölzl
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Introduce CommRing -- the category of commutative rings.
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Currently only the basic setup.
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-/
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import category_theory.instances.monoids
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import category_theory.fully_faithful
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import category_theory.adjunction
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import data.mv_polynomial
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import algebra.ring
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import data.int.basic
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universes u v
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⟨by introsI α ia; apply_instance,
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by introsI α β γ ia ib ic f g hf hg; apply_instance⟩
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instance Ring_hom_is_ring_hom {R S : Ring} (f : R ⟶ S) : is_ring_hom (f : R → S) := f.2
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instance Ring.hom_is_ring_hom {R S : Ring} (f : R ⟶ S) : is_ring_hom (f : R → S) := f.2
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/-- The category of commutative rings. -/
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@[reducible] def CommRing : Type (u+1) := bundled comm_ring
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example : faithful (forget_to_CommMon ⋙ CommMon.forget_to_Mon) := by apply_instance
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section
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open mv_polynomial
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local attribute [instance, priority 0] subtype.fintype set_fintype classical.prop_decidable
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noncomputable def polynomial : Type u ⥤ CommRing.{u} :=
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{ obj := λ α, ⟨mv_polynomial α ℤ, by apply_instance⟩,
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map := λ α β f, ⟨eval₂ C (X ∘ f), by apply_instance⟩,
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map_id' := λ α, subtype.ext.mpr $ funext $ eval₂_eta,
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map_comp' := λ α β γ f g, subtype.ext.mpr $ funext $ λ p,
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by apply mv_polynomial.induction_on p; intros;
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simp only [*, eval₂_add, eval₂_mul, eval₂_C, eval₂_X, comp_val,
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eq_self_iff_true, function.comp_app, types_comp] at * }
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@[simp] lemma polynomial_obj_α {α : Type u} :
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(polynomial.obj α).α = mv_polynomial α ℤ := rfl
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@[simp] lemma polynomial_map_val {α β : Type u} {f : α → β} :
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(CommRing.polynomial.map f).val = eval₂ C (X ∘ f) := rfl
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noncomputable def adj : adjunction polynomial (forget : CommRing ⥤ Type u) :=
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adjunction.mk_of_hom_equiv _ _
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{ hom_equiv := λ α R,
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{ to_fun := λ f, f ∘ X,
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inv_fun := λ f, ⟨eval₂ int.cast f, by apply_instance⟩,
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left_inv := λ f, subtype.ext.mpr $ funext $ λ p,
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begin
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have H0 := λ n, (congr (int.eq_cast' (f.val ∘ C)) (rfl : n = n)).symm,
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have H1 := λ p₁ p₂, (@is_ring_hom.map_add _ _ _ _ f.val f.2 p₁ p₂).symm,
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have H2 := λ p₁ p₂, (@is_ring_hom.map_mul _ _ _ _ f.val f.2 p₁ p₂).symm,
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apply mv_polynomial.induction_on p; intros;
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simp only [*, eval₂_add, eval₂_mul, eval₂_C, eval₂_X,
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eq_self_iff_true, function.comp_app, hom_coe_app] at *
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end,
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right_inv := by tidy },
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hom_equiv_naturality_left_symm' := λ X' X Y f g, subtype.ext.mpr $ funext $ λ p,
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begin
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apply mv_polynomial.induction_on p; intros;
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simp only [*, eval₂_mul, eval₂_add, eval₂_C, eval₂_X,
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comp_val, equiv.coe_fn_symm_mk, hom_coe_app, polynomial_map_val,
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eq_self_iff_true, function.comp_app, add_right_inj, types_comp] at *
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end }
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end
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end CommRing
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end category_theory.instances

‎src/category_theory/instances/CommRing/default.lean

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/-
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Copyright (c) 2019 Scott Morrison. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Scott Morrison
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-/
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import category_theory.instances.CommRing.basic
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import category_theory.instances.CommRing.adjunctions

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