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Mon.forget_reflects_isos : reflects_isomorphisms (forget Mon.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget Mon).map f),
let e : X ≃* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_Mon_iso).1⟩,
end } | instance | Mon.forget_reflects_isos | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommMon.forget_reflects_isos : reflects_isomorphisms (forget CommMon.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget CommMon).map f),
let e : X ≃* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_CommMon_iso).1⟩,
end } | instance | CommMon.forget_reflects_isos | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prequotient
-- There's always `of`
| of : Π (j : J) (x : F.obj j), prequotient
-- Then one generator for each operation
| one : prequotient
| mul : prequotient → prequotient → prequotient | inductive | Mon.colimits.prequotient | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | An inductive type representing all monoid expressions (without relations)
on a collection of types indexed by the objects of `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relation : prequotient F → prequotient F → Prop
-- Make it an equivalence relation:
| refl : Π (x), relation x x
| symm : Π (x y) (h : relation x y), relation y x
| trans : Π (x y z) (h : relation x y) (k : relation y z), relation x z
-- There's always a `map` relation
| map : Π (j j' : J) (f : j ⟶ j') (x : F.obj j), r... | inductive | Mon.colimits.relation | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"mul_assoc",
"mul_one",
"one_mul"
] | The relation on `prequotient` saying when two expressions are equal
because of the monoid laws, or
because one element is mapped to another by a morphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_colimit_type : monoid (colimit_type F) | { mul :=
begin
fapply @quot.lift _ _ ((colimit_type F) → (colimit_type F)),
{ intro x,
fapply @quot.lift,
{ intro y,
exact quot.mk _ (mul x y) },
{ intros y y' r,
apply quot.sound,
exact relation.mul_2 _ _ _ r } },
{ intros x x' r,
funext y,
induction ... | instance | Mon.colimits.monoid_colimit_type | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"monoid",
"mul_assoc",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_one : quot.mk setoid.r one = (1 : colimit_type F) | rfl | lemma | Mon.colimits.quot_one | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_mul (x y) : quot.mk setoid.r (mul x y) =
((quot.mk setoid.r x) * (quot.mk setoid.r y) : colimit_type F) | rfl | lemma | Mon.colimits.quot_mul | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : Mon | ⟨colimit_type F, by apply_instance⟩ | def | Mon.colimits.colimit | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"Mon"
] | The bundled monoid giving the colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (j : J) : F.obj j ⟶ colimit F | { to_fun := cocone_fun F j,
map_one' := quot.sound (relation.one _),
map_mul' := λ x y, quot.sound (relation.mul _ _ _) } | def | Mon.colimits.cocone_morphism | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The monoid homomorphism from a given monoid in the diagram to the colimit monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit F,
ι :=
{ app := cocone_morphism F, } }. | def | Mon.colimits.colimit_cocone | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The cocone over the proposed colimit monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_fun_lift (s : cocone F) : prequotient F → s.X | | (of j x) := (s.ι.app j) x
| one := 1
| (mul x y) := desc_fun_lift x * desc_fun_lift y | def | Mon.colimits.desc_fun_lift | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The function from the free monoid on the diagram to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_fun (s : cocone F) : colimit_type F → s.X | begin
fapply quot.lift,
{ exact desc_fun_lift F s },
{ intros x y r,
induction r; try { dsimp },
-- refl
{ refl },
-- symm
{ exact r_ih.symm },
-- trans
{ exact eq.trans r_ih_h r_ih_k },
-- map
{ simp, },
-- mul
{ simp, },
-- one
{ simp, },
-- mul_1
{ rw... | def | Mon.colimits.desc_fun | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"mul_assoc",
"mul_one",
"one_mul"
] | The function from the colimit monoid to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_morphism (s : cocone F) : colimit F ⟶ s.X | { to_fun := desc_fun F s,
map_one' := rfl,
map_mul' := λ x y, by { induction x; induction y; refl }, } | def | Mon.colimits.desc_morphism | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The monoid homomorphism from the colimit monoid to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_is_colimit : is_colimit (colimit_cocone F) | { desc := λ s, desc_morphism F s,
uniq' := λ s m w,
begin
ext,
induction x,
induction x,
{ have w' := congr_fun (congr_arg (λ f : F.obj x_j ⟶ s.X, (f : F.obj x_j → s.X)) (w x_j)) x_x,
erw w',
refl, },
{ simp *, },
{ simp *, },
refl
end }. | def | Mon.colimits.colimit_is_colimit | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | Evidence that the proposed colimit is the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_Mon : has_colimits Mon | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk
{ cocone := colimit_cocone F,
is_colimit := colimit_is_colimit F } } } | instance | Mon.colimits.has_colimits_Mon | algebra.category.Mon | src/algebra/category/Mon/colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
M : Type (max v u) | types.quot (F ⋙ forget Mon) | abbreviation | Mon.filtered_colimits.M | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon"
] | The colimit of `F ⋙ forget Mon` in the category of types.
In the following, we will construct a monoid structure on `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
M.mk : (Σ j, F.obj j) → M | quot.mk (types.quot.rel (F ⋙ forget Mon)) | abbreviation | Mon.filtered_colimits.M.mk | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon"
] | The canonical projection into the colimit, as a quotient type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
M.mk_eq (x y : Σ j, F.obj j)
(h : ∃ (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) :
M.mk x = M.mk y | quot.eqv_gen_sound (types.filtered_colimit.eqv_gen_quot_rel_of_rel (F ⋙ forget Mon) x y h) | lemma | Mon.filtered_colimits.M.mk_eq | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_has_one : has_one M | { one := M.mk ⟨is_filtered.nonempty.some, 1⟩ } | instance | Mon.filtered_colimits.colimit_has_one | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [] | As `J` is nonempty, we can pick an arbitrary object `j₀ : J`. We use this object to define the
"one" in the colimit as the equivalence class of `⟨j₀, 1 : F.obj j₀⟩`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_one_eq (j : J) : (1 : M) = M.mk ⟨j, 1⟩ | begin
apply M.mk_eq,
refine ⟨max' _ j, left_to_max _ j, right_to_max _ j, _⟩,
simp,
end | lemma | Mon.filtered_colimits.colimit_one_eq | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [] | The definition of the "one" in the colimit is independent of the chosen object of `J`.
In particular, this lemma allows us to "unfold" the definition of `colimit_one` at a custom chosen
object `j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_mul_aux (x y : Σ j, F.obj j) : M | M.mk ⟨max' x.1 y.1, F.map (left_to_max x.1 y.1) x.2 * F.map (right_to_max x.1 y.1) y.2⟩ | def | Mon.filtered_colimits.colimit_mul_aux | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [] | The "unlifted" version of multiplication in the colimit. To multiply two dependent pairs
`⟨j₁, x⟩` and `⟨j₂, y⟩`, we pass to a common successor of `j₁` and `j₂` (given by `is_filtered.max`)
and multiply them there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_mul_aux_eq_of_rel_left {x x' y : Σ j, F.obj j}
(hxx' : types.filtered_colimit.rel (F ⋙ forget Mon) x x') :
colimit_mul_aux x y = colimit_mul_aux x' y | begin
cases x with j₁ x, cases y with j₂ y, cases x' with j₃ x',
obtain ⟨l, f, g, hfg⟩ := hxx',
simp at hfg,
obtain ⟨s, α, β, γ, h₁, h₂, h₃⟩ := tulip (left_to_max j₁ j₂) (right_to_max j₁ j₂)
(right_to_max j₃ j₂) (left_to_max j₃ j₂) f g,
apply M.mk_eq,
use [s, α, γ],
dsimp,
simp_rw [monoid_hom.map_mu... | lemma | Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_left | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon",
"monoid_hom.map_mul"
] | Multiplication in the colimit is well-defined in the left argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_mul_aux_eq_of_rel_right {x y y' : Σ j, F.obj j}
(hyy' : types.filtered_colimit.rel (F ⋙ forget Mon) y y') :
colimit_mul_aux x y = colimit_mul_aux x y' | begin
cases y with j₁ y, cases x with j₂ x, cases y' with j₃ y',
obtain ⟨l, f, g, hfg⟩ := hyy',
simp at hfg,
obtain ⟨s, α, β, γ, h₁, h₂, h₃⟩ := tulip (right_to_max j₂ j₁) (left_to_max j₂ j₁)
(left_to_max j₂ j₃) (right_to_max j₂ j₃) f g,
apply M.mk_eq,
use [s, α, γ],
dsimp,
simp_rw [monoid_hom.map_mu... | lemma | Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_right | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon",
"monoid_hom.map_mul"
] | Multiplication in the colimit is well-defined in the right argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_has_mul : has_mul M | { mul := λ x y, begin
refine quot.lift₂ (colimit_mul_aux F) _ _ x y,
{ intros x y y' h,
apply colimit_mul_aux_eq_of_rel_right,
apply types.filtered_colimit.rel_of_quot_rel,
exact h },
{ intros x x' y h,
apply colimit_mul_aux_eq_of_rel_left,
apply types.filtered_colimit.rel_of_quot_rel,
e... | instance | Mon.filtered_colimits.colimit_has_mul | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"quot.lift₂"
] | Multiplication in the colimit. See also `colimit_mul_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_mul_mk_eq (x y : Σ j, F.obj j) (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k) :
(M.mk x) * (M.mk y) = M.mk ⟨k, F.map f x.2 * F.map g y.2⟩ | begin
cases x with j₁ x, cases y with j₂ y,
obtain ⟨s, α, β, h₁, h₂⟩ := bowtie (left_to_max j₁ j₂) f (right_to_max j₁ j₂) g,
apply M.mk_eq,
use [s, α, β],
dsimp,
simp_rw [monoid_hom.map_mul, ← comp_apply, ← F.map_comp, h₁, h₂],
end | lemma | Mon.filtered_colimits.colimit_mul_mk_eq | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"monoid_hom.map_mul"
] | Multiplication in the colimit is independent of the chosen "maximum" in the filtered category.
In particular, this lemma allows us to "unfold" the definition of the multiplication of `x` and `y`,
using a custom object `k` and morphisms `f : x.1 ⟶ k` and `g : y.1 ⟶ k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_monoid : monoid M | { one_mul := λ x, begin
apply quot.induction_on x, clear x, intro x, cases x with j x,
rw [colimit_one_eq F j, colimit_mul_mk_eq F ⟨j, 1⟩ ⟨j, x⟩ j (𝟙 j) (𝟙 j),
monoid_hom.map_one, one_mul, F.map_id, id_apply],
end,
mul_one := λ x, begin
apply quot.induction_on x, clear x, intro x, cases x with j... | instance | Mon.filtered_colimits.colimit_monoid | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"monoid",
"monoid_hom.map_one",
"mul_assoc",
"mul_one",
"one_mul",
"quot.induction_on₃"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : Mon | Mon.of M | def | Mon.filtered_colimits.colimit | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon",
"Mon.of"
] | The bundled monoid giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (j : J) : F.obj j ⟶ colimit | { to_fun := (types.colimit_cocone (F ⋙ forget Mon)).ι.app j,
map_one' := (colimit_one_eq j).symm,
map_mul' := λ x y, begin
convert (colimit_mul_mk_eq F ⟨j, x⟩ ⟨j, y⟩ j (𝟙 j) (𝟙 j)).symm,
rw [F.map_id, id_apply, id_apply], refl,
end } | def | Mon.filtered_colimits.cocone_morphism | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon"
] | The monoid homomorphism from a given monoid in the diagram to the colimit monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_naturality {j j' : J} (f : j ⟶ j') :
F.map f ≫ (cocone_morphism j') = cocone_morphism j | monoid_hom.coe_inj ((types.colimit_cocone (F ⋙ forget Mon)).ι.naturality f) | lemma | Mon.filtered_colimits.cocone_naturality | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon",
"monoid_hom.coe_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone : cocone F | { X := colimit,
ι := { app := cocone_morphism } }. | def | Mon.filtered_colimits.colimit_cocone | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [] | The cocone over the proposed colimit monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_desc (t : cocone F) : colimit ⟶ t.X | { to_fun := (types.colimit_cocone_is_colimit (F ⋙ forget Mon)).desc ((forget Mon).map_cocone t),
map_one' := begin
rw colimit_one_eq F is_filtered.nonempty.some,
exact monoid_hom.map_one _,
end,
map_mul' := λ x y, begin
apply quot.induction_on₂ x y, clear x y, intros x y,
cases x with i x, cases y... | def | Mon.filtered_colimits.colimit_desc | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon",
"monoid_hom.map_mul",
"monoid_hom.map_one",
"quot.induction_on₂"
] | Given a cocone `t` of `F`, the induced monoid homomorphism from the colimit to the cocone point.
As a function, this is simply given by the induced map of the corresponding cocone in `Type`.
The only thing left to see is that it is a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := colimit_desc,
fac' := λ t j, monoid_hom.coe_inj
((types.colimit_cocone_is_colimit (F ⋙ forget Mon)).fac ((forget Mon).map_cocone t) j),
uniq' := λ t m h, monoid_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget Mon)).uniq ((forget Mon).map_cocone t) m
(λ j, funext $ λ x, monoid_hom.... | def | Mon.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"Mon",
"monoid_hom.coe_inj",
"monoid_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_filtered_colimits : preserves_filtered_colimits (forget Mon.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit.{u u} F) (types.colimit_cocone_is_colimit (F ⋙ forget Mon.{u})) } } | instance | Mon.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
M : Mon | Mon.filtered_colimits.colimit (F ⋙ forget₂ CommMon Mon.{max v u}) | abbreviation | CommMon.filtered_colimits.M | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"CommMon",
"Mon",
"Mon.filtered_colimits.colimit"
] | The colimit of `F ⋙ forget₂ CommMon Mon` in the category `Mon`.
In the following, we will show that this has the structure of a _commutative_ monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_comm_monoid : comm_monoid M | { mul_comm := λ x y, begin
apply quot.induction_on₂ x y, clear x y, intros x y,
let k := max' x.1 y.1,
let f := left_to_max x.1 y.1,
let g := right_to_max x.1 y.1,
rw [colimit_mul_mk_eq _ x y k f g, colimit_mul_mk_eq _ y x k g f],
dsimp,
rw mul_comm,
end
..M.monoid } | instance | CommMon.filtered_colimits.colimit_comm_monoid | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"comm_monoid",
"mul_comm",
"quot.induction_on₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : CommMon | CommMon.of M | def | CommMon.filtered_colimits.colimit | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"CommMon",
"CommMon.of"
] | The bundled commutative monoid giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι := { ..(Mon.filtered_colimits.colimit_cocone (F ⋙ forget₂ CommMon Mon.{max v u})).ι } } | def | CommMon.filtered_colimits.colimit_cocone | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"CommMon",
"Mon.filtered_colimits.colimit_cocone"
] | The cocone over the proposed colimit commutative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t, Mon.filtered_colimits.colimit_desc (F ⋙ forget₂ CommMon Mon.{max v u})
((forget₂ CommMon Mon.{max v u}).map_cocone t),
fac' := λ t j, monoid_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget CommMon)).fac ((forget CommMon).map_cocone t) j,
uniq' := λ t m h, monoid_hom.coe_inj $
(... | def | CommMon.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"CommMon",
"Mon.filtered_colimits.colimit_desc",
"monoid_hom.coe_inj",
"monoid_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `CommMon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Mon_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ CommMon Mon.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit.{u u} F)
(Mon.filtered_colimits.colimit_cocone_is_colimit (F ⋙ forget₂ CommMon Mon.{u})) } } | instance | CommMon.filtered_colimits.forget₂_Mon_preserves_filtered_colimits | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"CommMon",
"Mon.filtered_colimits.colimit_cocone_is_colimit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget CommMon.{u}) | limits.comp_preserves_filtered_colimits (forget₂ CommMon Mon) (forget Mon) | instance | CommMon.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Mon | src/algebra/category/Mon/filtered_colimits.lean | [
"algebra.category.Mon.basic",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.elementwise",
"category_theory.limits.types"
] | [
"CommMon",
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_obj (F : J ⥤ Mon.{max v u}) (j) :
monoid ((F ⋙ forget Mon).obj j) | by { change monoid (F.obj j), apply_instance } | instance | Mon.monoid_obj | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"Mon",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_submonoid (F : J ⥤ Mon.{max v u}) :
submonoid (Π j, F.obj j) | { carrier := (F ⋙ forget Mon).sections,
one_mem' := λ j j' f, by simp,
mul_mem' := λ a b ah bh j j' f,
begin
simp only [forget_map_eq_coe, functor.comp_map, monoid_hom.map_mul, pi.mul_apply],
dsimp [functor.sections] at ah bh,
rw [ah f, bh f],
end } | def | Mon.sections_submonoid | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"Mon",
"monoid_hom.map_mul",
"pi.mul_apply",
"submonoid"
] | The flat sections of a functor into `Mon` form a submonoid of all sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_monoid (F : J ⥤ Mon.{max v u}) :
monoid (types.limit_cone (F ⋙ forget Mon.{max v u})).X | (sections_submonoid F).to_monoid | instance | Mon.limit_monoid | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_π_monoid_hom (F : J ⥤ Mon.{max v u}) (j) :
(types.limit_cone (F ⋙ forget Mon)).X →* (F ⋙ forget Mon).obj j | { to_fun := (types.limit_cone (F ⋙ forget Mon)).π.app j,
map_one' := rfl,
map_mul' := λ x y, rfl } | def | Mon.limit_π_monoid_hom | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"Mon"
] | `limit.π (F ⋙ forget Mon) j` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ Mon.{max v u}) : cone F | { X := Mon.of (types.limit_cone (F ⋙ forget _)).X,
π :=
{ app := limit_π_monoid_hom F,
naturality' := λ j j' f,
monoid_hom.coe_inj ((types.limit_cone (F ⋙ forget _)).π.naturality f) } } | def | Mon.has_limits.limit_cone | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"Mon.of",
"monoid_hom.coe_inj"
] | Construction of a limit cone in `Mon`.
(Internal use only; use the limits API.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ Mon.{max v u}) : is_limit (limit_cone F) | begin
refine is_limit.of_faithful
(forget Mon) (types.limit_cone_is_limit _)
(λ s, ⟨_, _, _⟩) (λ s, rfl); tidy,
end | def | Mon.has_limits.limit_cone_is_limit | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"Mon"
] | Witness that the limit cone in `Mon` is a limit cone.
(Internal use only; use the limits API.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v} Mon.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit.mk
{ cone := limit_cone F,
is_limit := limit_cone_is_limit F } } } | instance | Mon.has_limits_of_size | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | The category of monoids has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits Mon.{u} | Mon.has_limits_of_size.{u u} | instance | Mon.has_limits | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size : preserves_limits_of_size.{v} (forget Mon.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (types.limit_cone_is_limit (F ⋙ forget _)) } } | instance | Mon.forget_preserves_limits_of_size | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | The forgetful functor from monoids to types preserves all limits.
This means the underlying type of a limit can be computed as a limit in the category of types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget Mon.{u}) | Mon.forget_preserves_limits_of_size.{u u} | instance | Mon.forget_preserves_limits | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid_obj (F : J ⥤ CommMon.{max v u}) (j) :
comm_monoid ((F ⋙ forget CommMon).obj j) | by { change comm_monoid (F.obj j), apply_instance } | instance | CommMon.comm_monoid_obj | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon",
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_comm_monoid (F : J ⥤ CommMon.{max v u}) :
comm_monoid (types.limit_cone (F ⋙ forget CommMon.{max v u})).X | @submonoid.to_comm_monoid (Π j, F.obj j) _
(Mon.sections_submonoid (F ⋙ forget₂ CommMon Mon.{max v u})) | instance | CommMon.limit_comm_monoid | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon",
"Mon.sections_submonoid",
"comm_monoid",
"submonoid.to_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone (F : J ⥤ CommMon.{max v u}) : cone F | lift_limit (limit.is_limit (F ⋙ (forget₂ CommMon Mon.{max v u}))) | def | CommMon.limit_cone | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon"
] | A choice of limit cone for a functor into `CommMon`.
(Generally, you'll just want to use `limit F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ CommMon.{max v u}) : is_limit (limit_cone F) | lifted_limit_is_limit _ | def | CommMon.limit_cone_is_limit | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | The chosen cone is a limit cone.
(Generally, you'll just want to use `limit.cone F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v v} CommMon.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit_of_created F (forget₂ CommMon Mon.{max v u}) } } | instance | CommMon.has_limits_of_size | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon"
] | The category of commutative monoids has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits CommMon.{u} | CommMon.has_limits_of_size.{u u} | instance | CommMon.has_limits | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_Mon_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ CommMon Mon.{max v u}) | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ F, by apply_instance } } | instance | CommMon.forget₂_Mon_preserves_limits_of_size | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon"
] | The forgetful functor from commutative monoids to monoids preserves all limits.
This means the underlying type of a limit can be computed as a limit in the category of monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Mon_preserves_limits : preserves_limits (forget₂ CommMon Mon.{u}) | CommMon.forget₂_Mon_preserves_limits_of_size.{u u} | instance | CommMon.forget₂_Mon_preserves_limits | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget CommMon.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, limits.comp_preserves_limit (forget₂ CommMon Mon) (forget Mon) } } | instance | CommMon.forget_preserves_limits_of_size | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [
"CommMon",
"Mon"
] | The forgetful functor from commutative monoids to types preserves all limits.
This means the underlying type of a limit can be computed as a limit in the category of types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget CommMon.{u}) | CommMon.forget_preserves_limits_of_size.{u u} | instance | CommMon.forget_preserves_limits | algebra.category.Mon | src/algebra/category/Mon/limits.lean | [
"algebra.category.Mon.basic",
"algebra.group.pi",
"category_theory.limits.creates",
"category_theory.limits.types",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free : Type u ⥤ CommRing.{u} | { obj := λ α, of (mv_polynomial α ℤ),
map := λ X Y f,
(↑(rename f : _ →ₐ[ℤ] _) : (mv_polynomial X ℤ →+* mv_polynomial Y ℤ)),
-- TODO these next two fields can be done by `tidy`, but the calls in `dsimp` and `simp` it
-- generates are too slow.
map_id' := λ X, ring_hom.ext $ rename_id,
map_comp' := λ X Y Z... | def | CommRing.free | algebra.category.Ring | src/algebra/category/Ring/adjunctions.lean | [
"algebra.category.Ring.basic",
"data.mv_polynomial.comm_ring"
] | [
"free",
"mv_polynomial",
"ring_hom.ext"
] | The free functor `Type u ⥤ CommRing` sending a type `X` to the multivariable (commutative)
polynomials with variables `x : X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_obj_coe {α : Type u} :
(free.obj α : Type u) = mv_polynomial α ℤ | rfl | lemma | CommRing.free_obj_coe | algebra.category.Ring | src/algebra/category/Ring/adjunctions.lean | [
"algebra.category.Ring.basic",
"data.mv_polynomial.comm_ring"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_map_coe {α β : Type u} {f : α → β} :
⇑(free.map f) = rename f | rfl | lemma | CommRing.free_map_coe | algebra.category.Ring | src/algebra/category/Ring/adjunctions.lean | [
"algebra.category.Ring.basic",
"data.mv_polynomial.comm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adj : free ⊣ forget CommRing.{u} | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X R, hom_equiv,
hom_equiv_naturality_left_symm' :=
λ _ _ Y f g, ring_hom.ext $ λ x, eval₂_cast_comp f (int.cast_ring_hom Y) g x } | def | CommRing.adj | algebra.category.Ring | src/algebra/category/Ring/adjunctions.lean | [
"algebra.category.Ring.basic",
"data.mv_polynomial.comm_ring"
] | [
"adj",
"free",
"int.cast_ring_hom",
"ring_hom.ext"
] | The free-forgetful adjunction for commutative rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
SemiRing : Type (u+1) | bundled semiring | def | SemiRing | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"semiring"
] | The category of semirings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
assoc_ring_hom (M N : Type*) [semiring M] [semiring N] | ring_hom M N | abbreviation | SemiRing.assoc_ring_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"ring_hom",
"semiring"
] | `ring_hom` doesn't actually assume associativity. This alias is needed to make the category
theory machinery work. We use the same trick in `category_theory.Mon.assoc_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom : bundled_hom assoc_ring_hom | ⟨λ M N [semiring M] [semiring N], by exactI @ring_hom.to_fun M N _ _,
λ M [semiring M], by exactI @ring_hom.id M _,
λ M N P [semiring M] [semiring N] [semiring P], by exactI @ring_hom.comp M N P _ _ _,
λ M N [semiring M] [semiring N], by exactI @ring_hom.coe_inj M N _ _⟩ | instance | SemiRing.bundled_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"ring_hom.coe_inj",
"ring_hom.comp",
"ring_hom.id",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (R : Type u) [semiring R] : SemiRing | bundled.of R | def | SemiRing.of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"SemiRing",
"semiring"
] | Construct a bundled SemiRing from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {R S : Type u} [semiring R] [semiring S] (f : R →+* S) : of R ⟶ of S | f | def | SemiRing.of_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"semiring"
] | Typecheck a `ring_hom` as a morphism in `SemiRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {R S : Type u} [semiring R] [semiring S] (f : R →+* S) (x : R) :
of_hom f x = f x | rfl | lemma | SemiRing.of_hom_apply | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [semiring R] : (SemiRing.of R : Type u) = R | rfl | lemma | SemiRing.coe_of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"SemiRing.of",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Mon : has_forget₂ SemiRing Mon | bundled_hom.mk_has_forget₂
(λ R hR, @monoid_with_zero.to_monoid R (@semiring.to_monoid_with_zero R hR))
(λ R₁ R₂, ring_hom.to_monoid_hom) (λ _ _ _, rfl) | instance | SemiRing.has_forget_to_Mon | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Mon",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_AddCommMon : has_forget₂ SemiRing AddCommMon | -- can't use bundled_hom.mk_has_forget₂, since AddCommMon is an induced category
{ forget₂ :=
{ obj := λ R, AddCommMon.of R,
map := λ R₁ R₂ f, ring_hom.to_add_monoid_hom f } } | instance | SemiRing.has_forget_to_AddCommMon | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ring : Type (u+1) | bundled ring | def | Ring | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"ring"
] | The category of rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (R : Type u) [ring R] : Ring | bundled.of R | def | Ring.of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring",
"ring"
] | Construct a bundled Ring from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {R S : Type u} [ring R] [ring S] (f : R →+* S) : of R ⟶ of S | f | def | Ring.of_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"ring"
] | Typecheck a `ring_hom` as a morphism in `Ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {R S : Type u} [ring R] [ring S] (f : R →+* S) (x : R) :
of_hom f x = f x | rfl | lemma | Ring.of_hom_apply | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [ring R] : (Ring.of R : Type u) = R | rfl | lemma | Ring.coe_of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring.of",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_SemiRing : has_forget₂ Ring SemiRing | bundled_hom.forget₂ _ _ | instance | Ring.has_forget_to_SemiRing | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_AddCommGroup : has_forget₂ Ring AddCommGroup | -- can't use bundled_hom.mk_has_forget₂, since AddCommGroup is an induced category
{ forget₂ :=
{ obj := λ R, AddCommGroup.of R,
map := λ R₁ R₂ f, ring_hom.to_add_monoid_hom f } } | instance | Ring.has_forget_to_AddCommGroup | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommSemiRing : Type (u+1) | bundled comm_semiring | def | CommSemiRing | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"comm_semiring"
] | The category of commutative semirings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (R : Type u) [comm_semiring R] : CommSemiRing | bundled.of R | def | CommSemiRing.of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommSemiRing",
"comm_semiring"
] | Construct a bundled CommSemiRing from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {R S : Type u} [comm_semiring R] [comm_semiring S] (f : R →+* S) : of R ⟶ of S | f | def | CommSemiRing.of_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"comm_semiring"
] | Typecheck a `ring_hom` as a morphism in `CommSemiRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {R S : Type u} [comm_semiring R] [comm_semiring S] (f : R →+* S) (x : R) :
of_hom f x = f x | rfl | lemma | CommSemiRing.of_hom_apply | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [comm_semiring R] : (CommSemiRing.of R : Type u) = R | rfl | lemma | CommSemiRing.coe_of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommSemiRing.of",
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_SemiRing : has_forget₂ CommSemiRing SemiRing | bundled_hom.forget₂ _ _ | instance | CommSemiRing.has_forget_to_SemiRing | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommSemiRing",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_CommMon : has_forget₂ CommSemiRing CommMon | has_forget₂.mk'
(λ R : CommSemiRing, CommMon.of R) (λ R, rfl)
(λ R₁ R₂ f, f.to_monoid_hom) (by tidy) | instance | CommSemiRing.has_forget_to_CommMon | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommMon",
"CommMon.of",
"CommSemiRing"
] | The forgetful functor from commutative rings to (multiplicative) commutative monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommRing : Type (u+1) | bundled comm_ring | def | CommRing | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"comm_ring"
] | The category of commutative rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (R : Type u) [comm_ring R] : CommRing | bundled.of R | def | CommRing.of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing",
"comm_ring"
] | Construct a bundled CommRing from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {R S : Type u} [comm_ring R] [comm_ring S] (f : R →+* S) : of R ⟶ of S | f | def | CommRing.of_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"comm_ring"
] | Typecheck a `ring_hom` as a morphism in `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {R S : Type u} [comm_ring R] [comm_ring S] (f : R →+* S) (x : R) :
of_hom f x = f x | rfl | lemma | CommRing.of_hom_apply | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [comm_ring R] : (CommRing.of R : Type u) = R | rfl | lemma | CommRing.coe_of | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing.of",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Ring : has_forget₂ CommRing Ring | bundled_hom.forget₂ _ _ | instance | CommRing.has_forget_to_Ring | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing",
"Ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_CommSemiRing : has_forget₂ CommRing CommSemiRing | has_forget₂.mk' (λ R : CommRing, CommSemiRing.of R) (λ R, rfl) (λ R₁ R₂ f, f) (by tidy) | instance | CommRing.has_forget_to_CommSemiRing | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing",
"CommSemiRing",
"CommSemiRing.of"
] | The forgetful functor from commutative rings to (multiplicative) commutative monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Ring_iso [ring X] [ring Y] (e : X ≃+* Y) : Ring.of X ≅ Ring.of Y | { hom := e.to_ring_hom,
inv := e.symm.to_ring_hom } | def | ring_equiv.to_Ring_iso | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring.of",
"ring"
] | Build an isomorphism in the category `Ring` from a `ring_equiv` between `ring`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_CommRing_iso [comm_ring X] [comm_ring Y] (e : X ≃+* Y) :
CommRing.of X ≅ CommRing.of Y | { hom := e.to_ring_hom,
inv := e.symm.to_ring_hom } | def | ring_equiv.to_CommRing_iso | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing.of",
"comm_ring"
] | Build an isomorphism in the category `CommRing` from a `ring_equiv` between `comm_ring`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ring_iso_to_ring_equiv {X Y : Ring} (i : X ≅ Y) : X ≃+* Y | { to_fun := i.hom,
inv_fun := i.inv,
left_inv := by tidy,
right_inv := by tidy,
map_add' := by tidy,
map_mul' := by tidy }. | def | category_theory.iso.Ring_iso_to_ring_equiv | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring",
"inv_fun"
] | Build a `ring_equiv` from an isomorphism in the category `Ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommRing_iso_to_ring_equiv {X Y : CommRing} (i : X ≅ Y) : X ≃+* Y | { to_fun := i.hom,
inv_fun := i.inv,
left_inv := by tidy,
right_inv := by tidy,
map_add' := by tidy,
map_mul' := by tidy }. | def | category_theory.iso.CommRing_iso_to_ring_equiv | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing",
"inv_fun"
] | Build a `ring_equiv` from an isomorphism in the category `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommRing_iso_to_ring_equiv_to_ring_hom {X Y : CommRing} (i : X ≅ Y) :
i.CommRing_iso_to_ring_equiv.to_ring_hom = i.hom | by { ext, refl } | lemma | category_theory.iso.CommRing_iso_to_ring_equiv_to_ring_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommRing_iso_to_ring_equiv_symm_to_ring_hom {X Y : CommRing} (i : X ≅ Y) :
i.CommRing_iso_to_ring_equiv.symm.to_ring_hom = i.inv | by { ext, refl } | lemma | category_theory.iso.CommRing_iso_to_ring_equiv_symm_to_ring_hom | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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