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alg_equiv.to_Algebra_iso
{g₁ : ring X₁} {g₂ : ring X₂} {m₁ : algebra R X₁} {m₂ : algebra R X₂} (e : X₁ ≃ₐ[R] X₂) :
Algebra.of R X₁ ≅ Algebra.of R X₂ | { hom := (e : X₁ →ₐ[R] X₂),
inv := (e.symm : X₂ →ₐ[R] X₁),
hom_inv_id' := begin ext, exact e.left_inv x, end,
inv_hom_id' := begin ext, exact e.right_inv x, end, } | def | alg_equiv.to_Algebra_iso | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"Algebra.of",
"algebra",
"ring"
] | Build an isomorphism in the category `Algebra R` from a `alg_equiv` between `algebra`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_alg_equiv {X Y : Algebra R} (i : X ≅ Y) : X ≃ₐ[R] 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,
commutes' := by tidy, }. | def | category_theory.iso.to_alg_equiv | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"Algebra",
"inv_fun"
] | Build a `alg_equiv` from an isomorphism in the category `Algebra R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_equiv_iso_Algebra_iso {X Y : Type u}
[ring X] [ring Y] [algebra R X] [algebra R Y] :
(X ≃ₐ[R] Y) ≅ (Algebra.of R X ≅ Algebra.of R Y) | { hom := λ e, e.to_Algebra_iso,
inv := λ i, i.to_alg_equiv, } | def | alg_equiv_iso_Algebra_iso | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"Algebra.of",
"algebra",
"ring"
] | Algebra equivalences between `algebras`s are the same as (isomorphic to) isomorphisms in
`Algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Algebra.forget_reflects_isos : reflects_isomorphisms (forget (Algebra.{u} R)) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget (Algebra.{u} R)).map f),
let e : X ≃ₐ[R] Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_Algebra_iso).1⟩,
end } | instance | Algebra.forget_reflects_isos | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring_obj (F : J ⥤ Algebra.{max v w} R) (j) :
semiring ((F ⋙ forget (Algebra R)).obj j) | by { change semiring (F.obj j), apply_instance } | instance | Algebra.semiring_obj | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_obj (F : J ⥤ Algebra.{max v w} R) (j) :
algebra R ((F ⋙ forget (Algebra R)).obj j) | by { change algebra R (F.obj j), apply_instance } | instance | Algebra.algebra_obj | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_subalgebra (F : J ⥤ Algebra.{max v w} R) :
subalgebra R (Π j, F.obj j) | { algebra_map_mem' := λ r j j' f, (F.map f).commutes r,
..SemiRing.sections_subsemiring
(F ⋙ forget₂ (Algebra R) Ring.{max v w} ⋙ forget₂ Ring SemiRing.{max v w}) } | def | Algebra.sections_subalgebra | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"Ring",
"SemiRing.sections_subsemiring",
"subalgebra"
] | The flat sections of a functor into `Algebra R` form a submodule of all sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_semiring (F : J ⥤ Algebra.{max v w} R) :
ring (types.limit_cone (F ⋙ forget (Algebra.{max v w} R))).X | begin
change ring (sections_subalgebra F),
apply_instance,
end | instance | Algebra.limit_semiring | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_algebra (F : J ⥤ Algebra.{max v w} R) :
algebra R (types.limit_cone (F ⋙ forget (Algebra.{max v w} R))).X | begin
have : algebra R (types.limit_cone (F ⋙ forget (Algebra.{max v w} R))).X
= algebra R (sections_subalgebra F), by refl,
rw this,
apply_instance,
end | instance | Algebra.limit_algebra | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_π_alg_hom (F : J ⥤ Algebra.{max v w} R) (j) :
(types.limit_cone (F ⋙ forget (Algebra R))).X →ₐ[R] (F ⋙ forget (Algebra.{max v w} R)).obj j | { commutes' := λ r, rfl,
..SemiRing.limit_π_ring_hom
(F ⋙ forget₂ (Algebra R) Ring.{max v w} ⋙ forget₂ Ring SemiRing.{max v w}) j } | def | Algebra.limit_π_alg_hom | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"Ring",
"SemiRing.limit_π_ring_hom"
] | `limit.π (F ⋙ forget (Algebra R)) j` as a `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ Algebra.{max v w} R) : cone F | { X := Algebra.of R (types.limit_cone (F ⋙ forget _)).X,
π :=
{ app := limit_π_alg_hom F,
naturality' := λ j j' f,
alg_hom.coe_fn_injective ((types.limit_cone (F ⋙ forget _)).π.naturality f) } } | def | Algebra.has_limits.limit_cone | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra.of",
"alg_hom.coe_fn_injective"
] | Construction of a limit cone in `Algebra R`.
(Internal use only; use the limits API.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ Algebra.{max v w} R) : is_limit (limit_cone F) | begin
refine is_limit.of_faithful
(forget (Algebra R)) (types.limit_cone_is_limit _)
(λ s, { .. }) (λ s, rfl),
{ simp only [forget_map_eq_coe, alg_hom.map_one, functor.map_cone_π_app], refl, },
{ intros x y, simp only [forget_map_eq_coe, alg_hom.map_mul, functor.map_cone_π_app], refl, },
{ simp only [fo... | def | Algebra.has_limits.limit_cone_is_limit | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"alg_hom.map_add",
"alg_hom.map_mul",
"alg_hom.map_one",
"alg_hom.map_zero"
] | Witness that the limit cone in `Algebra R` 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 v} (Algebra.{max v w} R) | { 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 | Algebra.has_limits_of_size | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [] | The category of R-algebras has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits (Algebra.{w} R) | Algebra.has_limits_of_size.{w w u} | instance | Algebra.has_limits | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_Ring_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ (Algebra R) Ring.{max v w}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F)
(by apply Ring.limit_cone_is_limit (F ⋙ forget₂ (Algebra R) Ring.{max v w})) } } | instance | Algebra.forget₂_Ring_preserves_limits_of_size | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"Ring.limit_cone_is_limit"
] | The forgetful functor from R-algebras to rings preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Ring_preserves_limits :
preserves_limits (forget₂ (Algebra R) Ring.{w}) | Algebra.forget₂_Ring_preserves_limits_of_size.{w w} | instance | Algebra.forget₂_Ring_preserves_limits | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_Module_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ (Algebra R) (Module.{max v w} R)) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F)
(by apply Module.has_limits.limit_cone_is_limit
(F ⋙ forget₂ (Algebra R) (Module.{max v w} R))) } } | instance | Algebra.forget₂_Module_preserves_limits_of_size | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra",
"Module.has_limits.limit_cone_is_limit"
] | The forgetful functor from R-algebras to R-modules preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Module_preserves_limits : preserves_limits (forget₂ (Algebra R) (Module.{w} R)) | Algebra.forget₂_Module_preserves_limits_of_size.{w w} | instance | Algebra.forget₂_Module_preserves_limits | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [
"Algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget (Algebra.{max v w} R)) | { 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 | Algebra.forget_preserves_limits_of_size | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [] | The forgetful functor from R-algebras to types preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget (Algebra.{w} R)) | Algebra.forget_preserves_limits_of_size.{w w} | instance | Algebra.forget_preserves_limits | algebra.category.Algebra | src/algebra/category/Algebra/limits.lean | [
"algebra.category.Algebra.basic",
"algebra.category.Module.limits",
"algebra.category.Ring.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fgModule | full_subcategory (λ (V : Module.{u} R), module.finite R V) | def | fgModule | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"module.finite"
] | Define `fgModule` as the subtype of `Module.{u} R` of finitely generated modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite (V : fgModule R) : module.finite R V.obj | V.property | instance | fgModule.finite | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule",
"finite",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (V : Type u) [add_comm_group V] [module R V] [module.finite R V] : fgModule R | ⟨Module.of R V, by { change module.finite R V, apply_instance }⟩ | def | fgModule.of | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"add_comm_group",
"fgModule",
"module",
"module.finite"
] | Lift an unbundled finitely generated module to `fgModule R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_to_linear_equiv {V W : fgModule R} (i : V ≅ W) : V.obj ≃ₗ[R] W.obj | ((forget₂ (fgModule.{u} R) (Module.{u} R)).map_iso i).to_linear_equiv | def | fgModule.iso_to_linear_equiv | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule"
] | Converts and isomorphism in the category `fgModule R` to a `linear_equiv` between the underlying
modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.linear_equiv.to_fgModule_iso
{V W : Type u} [add_comm_group V] [module R V] [module.finite R V]
[add_comm_group W] [module R W] [module.finite R W]
(e : V ≃ₗ[R] W) :
fgModule.of R V ≅ fgModule.of R W | { hom := e.to_linear_map,
inv := e.symm.to_linear_map,
hom_inv_id' := by {ext, exact e.left_inv x},
inv_hom_id' := by {ext, exact e.right_inv x} } | def | linear_equiv.to_fgModule_iso | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"add_comm_group",
"fgModule.of",
"module",
"module.finite"
] | Converts a `linear_equiv` to an isomorphism in the category `fgModule R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoidal_predicate_module_finite :
monoidal_category.monoidal_predicate (λ V : Module.{u} R, module.finite R V) | { prop_id' := module.finite.self R,
prop_tensor' := λ X Y hX hY, by exactI module.finite.tensor_product R X Y } | instance | fgModule.monoidal_predicate_module_finite | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"module.finite",
"module.finite.self",
"module.finite.tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_monoidal : monoidal_functor (fgModule R) (Module.{u} R) | monoidal_category.full_monoidal_subcategory_inclusion _ | def | fgModule.forget₂_monoidal | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule"
] | The forgetful functor `fgModule R ⥤ Module R` as a monoidal functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_monoidal_faithful : faithful (forget₂_monoidal R).to_functor | by { dsimp [forget₂_monoidal], apply_instance, } | instance | fgModule.forget₂_monoidal_faithful | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_monoidal_additive : (forget₂_monoidal R).to_functor.additive | by { dsimp [forget₂_monoidal], apply_instance, } | instance | fgModule.forget₂_monoidal_additive | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_monoidal_linear : (forget₂_monoidal R).to_functor.linear R | by { dsimp [forget₂_monoidal], apply_instance, } | instance | fgModule.forget₂_monoidal_linear | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.conj_eq_conj {V W : fgModule R} (i : V ≅ W) (f : End V) :
iso.conj i f = linear_equiv.conj (iso_to_linear_equiv i) f | rfl | lemma | fgModule.iso.conj_eq_conj | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule",
"linear_equiv.conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_predicate_module_finite :
monoidal_category.closed_predicate (λ V : Module.{u} K, module.finite K V) | { prop_ihom' := λ X Y hX hY, by exactI @module.finite.linear_map K X Y _ _ _ _ _ _ _ hX hY } | instance | fgModule.closed_predicate_module_finite | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"module.finite",
"module.finite.linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_obj : (ihom V).obj W = fgModule.of K (V.obj →ₗ[K] W.obj) | rfl | lemma | fgModule.ihom_obj | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fgModule_dual : fgModule K | ⟨Module.of K (module.dual K V.obj), subspace.module.dual.finite_dimensional⟩ | def | fgModule.fgModule_dual | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule",
"module.dual"
] | The dual module is the dual in the rigid monoidal category `fgModule K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fgModule_coevaluation : 𝟙_ (fgModule K) ⟶ V ⊗ (fgModule_dual K V) | by apply coevaluation K V.obj | def | fgModule.fgModule_coevaluation | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"coevaluation",
"fgModule"
] | The coevaluation map is defined in `linear_algebra.coevaluation`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fgModule_coevaluation_apply_one : fgModule_coevaluation K V (1 : K) =
∑ (i : basis.of_vector_space_index K V.obj),
(basis.of_vector_space K V.obj) i ⊗ₜ[K] (basis.of_vector_space K V.obj).coord i | by apply coevaluation_apply_one K V.obj | lemma | fgModule.fgModule_coevaluation_apply_one | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"basis.of_vector_space",
"basis.of_vector_space_index",
"coevaluation_apply_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fgModule_evaluation : (fgModule_dual K V) ⊗ V ⟶ 𝟙_ (fgModule K) | by apply contract_left K V.obj | def | fgModule.fgModule_evaluation | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"contract_left",
"fgModule"
] | The evaluation morphism is given by the contraction map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fgModule_evaluation_apply (f : (fgModule_dual K V).obj) (x : V.obj) :
(fgModule_evaluation K V) (f ⊗ₜ x) = f.to_fun x | by apply contract_left_apply f x | lemma | fgModule.fgModule_evaluation_apply | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"contract_left_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coevaluation_evaluation :
let V' : fgModule K | fgModule_dual K V in
(𝟙 V' ⊗ (fgModule_coevaluation K V)) ≫ (α_ V' V V').inv ≫ (fgModule_evaluation K V ⊗ 𝟙 V')
= (ρ_ V').hom ≫ (λ_ V').inv :=
by apply contract_left_assoc_coevaluation K V.obj | theorem | fgModule.coevaluation_evaluation | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"contract_left_assoc_coevaluation",
"fgModule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_coevaluation :
(fgModule_coevaluation K V ⊗ 𝟙 V)
≫ (α_ V (fgModule_dual K V) V).hom ≫ (𝟙 V ⊗ fgModule_evaluation K V)
= (λ_ V).hom ≫ (ρ_ V).inv | by apply contract_left_assoc_coevaluation' K V.obj | theorem | fgModule.evaluation_coevaluation | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"contract_left_assoc_coevaluation'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_pairing : exact_pairing V (fgModule_dual K V) | { coevaluation := fgModule_coevaluation K V,
evaluation := fgModule_evaluation K V,
coevaluation_evaluation' := coevaluation_evaluation K V,
evaluation_coevaluation' := evaluation_coevaluation K V } | instance | fgModule.exact_pairing | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"coevaluation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_dual : has_right_dual V | ⟨fgModule_dual K V⟩ | instance | fgModule.right_dual | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_rigid_category : right_rigid_category (fgModule K) | { } | instance | fgModule.right_rigid_category | algebra.category.fgModule | src/algebra/category/fgModule/basic.lean | [
"category_theory.monoidal.rigid.basic",
"category_theory.monoidal.subcategory",
"linear_algebra.coevaluation",
"linear_algebra.free_module.finite.matrix",
"algebra.category.Module.monoidal.closed"
] | [
"fgModule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_creates_limit (F : J ⥤ fgModule k) :
creates_limit F (forget₂ (fgModule k) (Module.{v} k)) | creates_limit_of_fully_faithful_of_iso
⟨(limit (F ⋙ forget₂ (fgModule k) (Module.{v} k)) : Module.{v} k), by apply_instance⟩
(iso.refl _) | def | fgModule.forget₂_creates_limit | algebra.category.fgModule | src/algebra/category/fgModule/limits.lean | [
"algebra.category.fgModule.basic",
"algebra.category.Module.limits",
"algebra.category.Module.products",
"algebra.category.Module.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"fgModule"
] | The forgetful functor from `fgModule k` to `Module k` creates all finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono (hf : mono f) : normal_mono f | equivalence_reflects_normal_mono (forget₂ (Module.{u} ℤ) AddCommGroup.{u}).inv $
Module.normal_mono _ infer_instance | def | AddCommGroup.normal_mono | algebra.category.Group | src/algebra/category/Group/abelian.lean | [
"algebra.category.Group.Z_Module_equivalence",
"algebra.category.Group.limits",
"algebra.category.Group.colimits",
"algebra.category.Module.abelian",
"category_theory.abelian.basic"
] | [
"Module.normal_mono"
] | In the category of abelian groups, every monomorphism is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi (hf : epi f) : normal_epi f | equivalence_reflects_normal_epi (forget₂ (Module.{u} ℤ) AddCommGroup.{u}).inv $
Module.normal_epi _ infer_instance | def | AddCommGroup.normal_epi | algebra.category.Group | src/algebra/category/Group/abelian.lean | [
"algebra.category.Group.Z_Module_equivalence",
"algebra.category.Group.limits",
"algebra.category.Group.colimits",
"algebra.category.Module.abelian",
"category_theory.abelian.basic"
] | [
"Module.normal_epi"
] | In the category of abelian groups, every epimorphism is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free : Type u ⥤ AddCommGroup | { obj := λ α, of (free_abelian_group α),
map := λ X Y, free_abelian_group.map,
map_id' := λ X, add_monoid_hom.ext free_abelian_group.map_id_apply,
map_comp' := λ X Y Z f g, add_monoid_hom.ext free_abelian_group.map_comp_apply, } | def | AddCommGroup.free | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"free",
"free_abelian_group",
"free_abelian_group.map",
"free_abelian_group.map_comp_apply",
"free_abelian_group.map_id_apply"
] | The free functor `Type u ⥤ AddCommGroup` sending a type `X` to the
free abelian group with generators `x : X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_obj_coe {α : Type u} :
(free.obj α : Type u) = (free_abelian_group α) | rfl | lemma | AddCommGroup.free_obj_coe | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"free_abelian_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_map_coe {α β : Type u} {f : α → β} (x : free_abelian_group α) :
(free.map f) x = f <$> x | rfl | lemma | AddCommGroup.free_map_coe | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"free_abelian_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adj : free ⊣ forget AddCommGroup.{u} | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X G, free_abelian_group.lift.symm,
hom_equiv_naturality_left_symm' :=
by { intros, ext, refl} } | def | AddCommGroup.adj | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"adj",
"free"
] | The free-forgetful adjunction for abelian groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free : Type u ⥤ Group | { obj := λ α, of (free_group α),
map := λ X Y, free_group.map,
map_id' := by { intros, ext1, refl },
map_comp' := by { intros, ext1, refl } } | def | Group.free | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"Group",
"free",
"free_group",
"free_group.map"
] | The free functor `Type u ⥤ Group` sending a type `X` to the free group with generators `x : X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj : free ⊣ forget Group.{u} | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X G, free_group.lift.symm,
hom_equiv_naturality_left_symm' := λ X Y G f g, by { ext1, refl } } | def | Group.adj | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"adj",
"free"
] | The free-forgetful adjunction for groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelianize : Group.{u} ⥤ CommGroup.{u} | { obj := λ G, { α := abelianization G, str := by apply_instance },
map := λ G H f, abelianization.lift ( { to_fun := λ x, abelianization.of (f x),
map_one' := by simp,
map_mul' := by simp } ),
map_id' := by { intros, simp only [monoid_hom.mk_coe, coe_id], ext1, refl },
map_comp' := by { intros, simp only [coe... | def | abelianize | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"abelianization",
"abelianization.lift",
"abelianization.of",
"monoid_hom.mk_coe"
] | The abelianization functor `Group ⥤ CommGroup` sending a group `G` to its abelianization `Gᵃᵇ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelianize_adj : abelianize ⊣ forget₂ CommGroup.{u} Group.{u} | adjunction.mk_of_hom_equiv
{ hom_equiv := λ G A, abelianization.lift.symm,
hom_equiv_naturality_left_symm' := λ G H A f g, by { ext1, refl } } | def | abelianize_adj | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"abelianize"
] | The abelianization-forgetful adjuction from `Group` to `CommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Mon.units : Mon.{u} ⥤ Group.{u} | { obj := λ R, Group.of Rˣ,
map := λ R S f, Group.of_hom $ units.map f,
map_id' := λ X, monoid_hom.ext (λ x, units.ext rfl),
map_comp' := λ X Y Z f g, monoid_hom.ext (λ x, units.ext rfl) } | def | Mon.units | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"Group.of",
"Group.of_hom",
"monoid_hom.ext",
"units.ext",
"units.map"
] | The functor taking a monoid to its subgroup of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Group.forget₂_Mon_adj : forget₂ Group Mon ⊣ Mon.units.{u} | { hom_equiv := λ X Y,
{ to_fun := λ f, monoid_hom.to_hom_units f,
inv_fun := λ f, (units.coe_hom Y).comp f,
left_inv := λ f, monoid_hom.ext $ λ _, rfl,
right_inv := λ f, monoid_hom.ext $ λ _, units.ext rfl },
unit :=
{ app := λ X, { ..(@to_units X _).to_monoid_hom },
naturality' := λ X Y f, monoid... | def | Group.forget₂_Mon_adj | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"Group",
"Mon",
"inv_fun",
"monoid_hom.ext",
"monoid_hom.to_hom_units",
"to_units",
"units.coe_hom",
"units.ext"
] | The forgetful-units adjunction between `Group` and `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommMon.units : CommMon.{u} ⥤ CommGroup.{u} | { obj := λ R, CommGroup.of Rˣ,
map := λ R S f, CommGroup.of_hom $ units.map f,
map_id' := λ X, monoid_hom.ext (λ x, units.ext rfl),
map_comp' := λ X Y Z f g, monoid_hom.ext (λ x, units.ext rfl) } | def | CommMon.units | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"CommGroup.of",
"CommGroup.of_hom",
"monoid_hom.ext",
"units.ext",
"units.map"
] | The functor taking a monoid to its subgroup of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommGroup.forget₂_CommMon_adj : forget₂ CommGroup CommMon ⊣ CommMon.units.{u} | { hom_equiv := λ X Y,
{ to_fun := λ f, monoid_hom.to_hom_units f,
inv_fun := λ f, (units.coe_hom Y).comp f,
left_inv := λ f, monoid_hom.ext $ λ _, rfl,
right_inv := λ f, monoid_hom.ext $ λ _, units.ext rfl },
unit :=
{ app := λ X, { ..(@to_units X _).to_monoid_hom },
naturality' := λ X Y f, monoid... | def | CommGroup.forget₂_CommMon_adj | algebra.category.Group | src/algebra/category/Group/adjunctions.lean | [
"algebra.category.Group.basic",
"group_theory.free_abelian_group"
] | [
"CommGroup",
"CommMon",
"inv_fun",
"monoid_hom.ext",
"monoid_hom.to_hom_units",
"to_units",
"units.coe_hom",
"units.ext"
] | The forgetful-units adjunction between `CommGroup` and `CommMon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Group : Type (u+1) | bundled group | def | Group | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"group"
] | The category of groups and group morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (X : Type u) [group X] : Group | bundled.of X | def | Group.of | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group",
"group"
] | Construct a bundled `Group` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {X Y : Type u} [group X] [group Y] (f : X →* Y) : of X ⟶ of Y | f | def | Group.of_hom | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"group"
] | Typecheck a `monoid_hom` as a morphism in `Group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {X Y : Type*} [group X] [group Y] (f : X →* Y) (x : X) :
of_hom f x = f x | rfl | lemma | Group.of_hom_apply | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [group R] : (Group.of R : Type u) = R | rfl | lemma | Group.coe_of | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group.of",
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_unique (G : Type*) [group G] [i : unique G] : unique (Group.of G) | i | instance | Group.of_unique | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group.of",
"group",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (G H : Group) (g : G) : (1 : G ⟶ H) g = 1 | rfl | lemma | Group.one_apply | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (G H : Group) (f₁ f₂ : G ⟶ H) (w : ∀ x, f₁ x = f₂ x) : f₁ = f₂ | by { ext1, apply w } | lemma | Group.ext | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Mon : has_forget₂ Group Mon | bundled_hom.forget₂ _ _ | instance | Group.has_forget_to_Mon | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group",
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommGroup : Type (u+1) | bundled comm_group | def | CommGroup | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"comm_group"
] | The category of commutative groups and group morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ab | AddCommGroup | abbreviation | Ab | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [] | `Ab` is an abbreviation for `AddCommGroup`, for the sake of mathematicians' sanity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (G : Type u) [comm_group G] : CommGroup | bundled.of G | def | CommGroup.of | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup",
"comm_group"
] | Construct a bundled `CommGroup` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {X Y : Type u} [comm_group X] [comm_group Y] (f : X →* Y) :
of X ⟶ of Y | f | def | CommGroup.of_hom | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"comm_group"
] | Typecheck a `monoid_hom` as a morphism in `CommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {X Y : Type*} [comm_group X] [comm_group Y] (f : X →* Y)
(x : X) : of_hom f x = f x | rfl | lemma | CommGroup.of_hom_apply | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group_instance (G : CommGroup) : comm_group G | G.str | instance | CommGroup.comm_group_instance | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup",
"comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [comm_group R] : (CommGroup.of R : Type u) = R | rfl | lemma | CommGroup.coe_of | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup.of",
"comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_unique (G : Type*) [comm_group G] [i : unique G] : unique (CommGroup.of G) | i | instance | CommGroup.of_unique | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup.of",
"comm_group",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (G H : CommGroup) (g : G) : (1 : G ⟶ H) g = 1 | rfl | lemma | CommGroup.one_apply | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (G H : CommGroup) (f₁ f₂ : G ⟶ H) (w : ∀ x, f₁ x = f₂ x) : f₁ = f₂ | by { ext1, apply w } | lemma | CommGroup.ext | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Group : has_forget₂ CommGroup Group | bundled_hom.forget₂ _ _ | instance | CommGroup.has_forget_to_Group | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup",
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_CommMon : has_forget₂ CommGroup CommMon | induced_category.has_forget₂ (λ G : CommGroup, CommMon.of G) | instance | CommGroup.has_forget_to_CommMon | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup",
"CommMon",
"CommMon.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_hom {G : AddCommGroup.{0}} (g : G) : (AddCommGroup.of ℤ) ⟶ G | zmultiples_hom G g | def | AddCommGroup.as_hom | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"zmultiples_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_hom_apply {G : AddCommGroup.{0}} (g : G) (i : ℤ) : (as_hom g) i = i • g | rfl | lemma | AddCommGroup.as_hom_apply | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_hom_injective {G : AddCommGroup.{0}} : function.injective (@as_hom G) | λ h k w, by convert congr_arg (λ k : (AddCommGroup.of ℤ) ⟶ G, (k : ℤ → G) (1 : ℤ)) w; simp | lemma | AddCommGroup.as_hom_injective | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_hom_ext
{G : AddCommGroup.{0}} (f g : (AddCommGroup.of ℤ) ⟶ G) (w : f (1 : ℤ) = g (1 : ℤ)) : f = g | add_monoid_hom.ext_int w | lemma | AddCommGroup.int_hom_ext | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"add_monoid_hom.ext_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_of_mono {G H : AddCommGroup.{0}} (f : G ⟶ H) [mono f] : function.injective f | λ g₁ g₂ h,
begin
have t0 : as_hom g₁ ≫ f = as_hom g₂ ≫ f :=
begin
ext,
simpa [as_hom_apply] using h,
end,
have t1 : as_hom g₁ = as_hom g₂ := (cancel_mono _).1 t0,
apply as_hom_injective t1,
end | lemma | AddCommGroup.injective_of_mono | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.to_Group_iso {X Y : Group} (e : X ≃* Y) : X ≅ Y | { hom := e.to_monoid_hom,
inv := e.symm.to_monoid_hom } | def | mul_equiv.to_Group_iso | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group"
] | Build an isomorphism in the category `Group` from a `mul_equiv` between `group`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.to_CommGroup_iso {X Y : CommGroup} (e : X ≃* Y) : X ≅ Y | { hom := e.to_monoid_hom,
inv := e.symm.to_monoid_hom } | def | mul_equiv.to_CommGroup_iso | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup"
] | Build an isomorphism in the category `CommGroup` from a `mul_equiv` between `comm_group`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Group_iso_to_mul_equiv {X Y : Group} (i : X ≅ Y) : X ≃* Y | i.hom.to_mul_equiv i.inv i.hom_inv_id i.inv_hom_id | def | category_theory.iso.Group_iso_to_mul_equiv | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group"
] | Build a `mul_equiv` from an isomorphism in the category `Group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommGroup_iso_to_mul_equiv {X Y : CommGroup} (i : X ≅ Y) : X ≃* Y | i.hom.to_mul_equiv i.inv i.hom_inv_id i.inv_hom_id | def | category_theory.iso.CommGroup_iso_to_mul_equiv | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup"
] | Build a `mul_equiv` from an isomorphism in the category `CommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_iso_Group_iso {X Y : Group.{u}} : (X ≃* Y) ≅ (X ≅ Y) | { hom := λ e, e.to_Group_iso,
inv := λ i, i.Group_iso_to_mul_equiv, } | def | mul_equiv_iso_Group_iso | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [] | multiplicative equivalences between `group`s are the same as (isomorphic to) isomorphisms
in `Group` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_iso_CommGroup_iso {X Y : CommGroup.{u}} : X ≃* Y ≅ (X ≅ Y) | { hom := λ e, e.to_CommGroup_iso,
inv := λ i, i.CommGroup_iso_to_mul_equiv, } | def | mul_equiv_iso_CommGroup_iso | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [] | multiplicative equivalences between `comm_group`s are the same as (isomorphic to) isomorphisms
in `CommGroup` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_perm {α : Type u} : Group.of (Aut α) ≅ Group.of (equiv.perm α) | { hom := ⟨λ g, g.to_equiv, (by tidy), (by tidy)⟩,
inv := ⟨λ g, g.to_iso, (by tidy), (by tidy)⟩ } | def | category_theory.Aut.iso_perm | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group.of",
"equiv.perm"
] | The (bundled) group of automorphisms of a type is isomorphic to the (bundled) group
of permutations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_perm {α : Type u} : Aut α ≃* equiv.perm α | iso_perm.Group_iso_to_mul_equiv | def | category_theory.Aut.mul_equiv_perm | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"equiv.perm"
] | The (unbundled) group of automorphisms of a type is `mul_equiv` to the (unbundled) group
of permutations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Group.forget_reflects_isos : reflects_isomorphisms (forget Group.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget Group).map f),
let e : X ≃* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_Group_iso).1⟩,
end } | instance | Group.forget_reflects_isos | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommGroup.forget_reflects_isos : reflects_isomorphisms (forget CommGroup.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget CommGroup).map f),
let e : X ≃* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_CommGroup_iso).1⟩,
end } | instance | CommGroup.forget_reflects_isos | algebra.category.Group | src/algebra/category/Group/basic.lean | [
"algebra.category.Mon.basic",
"category_theory.endomorphism"
] | [
"CommGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_limit_cone (G H : AddCommGroup.{u}) : limits.limit_cone (pair G H) | { cone :=
{ X := AddCommGroup.of (G × H),
π := { app := λ j, discrete.cases_on j
(λ j, walking_pair.cases_on j (add_monoid_hom.fst G H) (add_monoid_hom.snd G H)),
naturality' := by rintros ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩; refl, }},
is_limit :=
{ lift := λ s, add_monoid_hom.prod (s.π.app ⟨walking_pair.left⟩) (s.π... | def | AddCommGroup.binary_product_limit_cone | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [
"lift"
] | Construct limit data for a binary product in `AddCommGroup`, using `AddCommGroup.of (G × H)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_product_limit_cone_cone_π_app_left (G H : AddCommGroup.{u}) :
(binary_product_limit_cone G H).cone.π.app ⟨walking_pair.left⟩ = add_monoid_hom.fst G H | rfl | lemma | AddCommGroup.binary_product_limit_cone_cone_π_app_left | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_limit_cone_cone_π_app_right (G H : AddCommGroup.{u}) :
(binary_product_limit_cone G H).cone.π.app ⟨walking_pair.right⟩ = add_monoid_hom.snd G H | rfl | lemma | AddCommGroup.binary_product_limit_cone_cone_π_app_right | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod_iso_prod (G H : AddCommGroup.{u}) : (G ⊞ H : AddCommGroup) ≅ AddCommGroup.of (G × H) | is_limit.cone_point_unique_up_to_iso
(binary_biproduct.is_limit G H)
(binary_product_limit_cone G H).is_limit | def | AddCommGroup.biprod_iso_prod | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | We verify that the biproduct in AddCommGroup is isomorphic to
the cartesian product of the underlying types: | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod_iso_prod_inv_comp_fst (G H : AddCommGroup.{u}) :
(biprod_iso_prod G H).inv ≫ biprod.fst = add_monoid_hom.fst G H | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.left) | lemma | AddCommGroup.biprod_iso_prod_inv_comp_fst | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod_iso_prod_inv_comp_snd (G H : AddCommGroup.{u}) :
(biprod_iso_prod G H).inv ≫ biprod.snd = add_monoid_hom.snd G H | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.right) | lemma | AddCommGroup.biprod_iso_prod_inv_comp_snd | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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