statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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cokernel_iso_range_quotient {G H : Module.{v} R} (f : G ⟶ H) :
cokernel f ≅ Module.of R (H ⧸ f.range) | colimit.iso_colimit_cocone ⟨_, cokernel_is_colimit f⟩ | def | Module.cokernel_iso_range_quotient | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"Module.of"
] | The categorical cokernel of a morphism in `Module`
agrees with the usual module-theoretical quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_π_cokernel_iso_range_quotient_hom :
cokernel.π f ≫ (cokernel_iso_range_quotient f).hom = f.range.mkq | by { convert colimit.iso_colimit_cocone_ι_hom _ _; refl, } | lemma | Module.cokernel_π_cokernel_iso_range_quotient_hom | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_mkq_cokernel_iso_range_quotient_inv :
↿f.range.mkq ≫ (cokernel_iso_range_quotient f).inv = cokernel.π f | by { convert colimit.iso_colimit_cocone_ι_inv ⟨_, cokernel_is_colimit f⟩ _; refl, } | lemma | Module.range_mkq_cokernel_iso_range_quotient_inv | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_π_ext {M N : Module.{u} R} (f : M ⟶ N) {x y : N} (m : M) (w : x = y + f m) :
cokernel.π f x = cokernel.π f y | by { subst w, simp, } | lemma | Module.cokernel_π_ext | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group_obj (F : J ⥤ Module.{max v w} R) (j) :
add_comm_group ((F ⋙ forget (Module R)).obj j) | by { change add_comm_group (F.obj j), apply_instance } | instance | Module.add_comm_group_obj | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_obj (F : J ⥤ Module.{max v w} R) (j) :
module R ((F ⋙ forget (Module R)).obj j) | by { change module R (F.obj j), apply_instance } | instance | Module.module_obj | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_submodule (F : J ⥤ Module.{max v w} R) :
submodule R (Π j, F.obj j) | { carrier := (F ⋙ forget (Module R)).sections,
smul_mem' := λ r s sh j j' f,
begin
simp only [forget_map_eq_coe, functor.comp_map, pi.smul_apply, linear_map.map_smul],
dsimp [functor.sections] at sh,
rw sh f,
end,
..(AddGroup.sections_add_subgroup
(F ⋙ forget₂ (Module R) AddCommGroup.{max v w}... | def | Module.sections_submodule | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"linear_map.map_smul",
"pi.smul_apply",
"submodule"
] | The flat sections of a functor into `Module R` form a submodule of all sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_add_comm_monoid (F : J ⥤ Module R) :
add_comm_monoid (types.limit_cone (F ⋙ forget (Module.{max v w} R))).X | show add_comm_monoid (sections_submodule F), by apply_instance | instance | Module.limit_add_comm_monoid | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_add_comm_group (F : J ⥤ Module R) :
add_comm_group (types.limit_cone (F ⋙ forget (Module.{max v w} R))).X | show add_comm_group (sections_submodule F), by apply_instance | instance | Module.limit_add_comm_group | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_module (F : J ⥤ Module R) :
module R (types.limit_cone (F ⋙ forget (Module.{max v w} R))).X | show module R (sections_submodule F), by apply_instance | instance | Module.limit_module | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_π_linear_map (F : J ⥤ Module R) (j) :
(types.limit_cone (F ⋙ forget (Module.{max v w} R))).X →ₗ[R] (F ⋙ forget (Module R)).obj j | { to_fun := (types.limit_cone (F ⋙ forget (Module R))).π.app j,
map_smul' := λ x y, rfl,
map_add' := λ x y, rfl } | def | Module.limit_π_linear_map | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module"
] | `limit.π (F ⋙ forget Ring) j` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ Module.{max v w} R) : cone F | { X := Module.of R (types.limit_cone (F ⋙ forget _)).X,
π :=
{ app := limit_π_linear_map F,
naturality' := λ j j' f,
linear_map.coe_injective ((types.limit_cone (F ⋙ forget _)).π.naturality f) } } | def | Module.has_limits.limit_cone | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module.of",
"linear_map.coe_injective"
] | Construction of a limit cone in `Module R`.
(Internal use only; use the limits API.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ Module.{max v w} R) : is_limit (limit_cone F) | by refine is_limit.of_faithful
(forget (Module R)) (types.limit_cone_is_limit _)
(λ s, ⟨_, _, _⟩) (λ s, rfl);
intros;
ext j;
simp only [subtype.coe_mk, functor.map_cone_π_app, forget_map_eq_coe,
linear_map.map_add, linear_map.map_smul];
refl | def | Module.has_limits.limit_cone_is_limit | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"linear_map.map_add",
"linear_map.map_smul",
"subtype.coe_mk"
] | Witness that the limit cone in `Module 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} (Module.{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 | Module.has_limits_of_size | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [] | The category of R-modules has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits (Module.{w} R) | Module.has_limits_of_size.{w w u} | instance | Module.has_limits | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_AddCommGroup_preserves_limits_aux (F : J ⥤ Module.{max v w} R) :
is_limit ((forget₂ (Module R) AddCommGroup).map_cone (limit_cone F)) | AddCommGroup.limit_cone_is_limit (F ⋙ forget₂ (Module R) AddCommGroup.{max v w}) | def | Module.forget₂_AddCommGroup_preserves_limits_aux | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module"
] | An auxiliary declaration to speed up typechecking. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommGroup_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ (Module R) AddCommGroup.{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) (forget₂_AddCommGroup_preserves_limits_aux F) } } | instance | Module.forget₂_AddCommGroup_preserves_limits_of_size | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module"
] | The forgetful functor from R-modules to abelian groups preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommGroup_preserves_limits :
preserves_limits (forget₂ (Module R) AddCommGroup.{w}) | Module.forget₂_AddCommGroup_preserves_limits_of_size.{w w} | instance | Module.forget₂_AddCommGroup_preserves_limits | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget (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) (types.limit_cone_is_limit (F ⋙ forget _)) } } | instance | Module.forget_preserves_limits_of_size | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [] | The forgetful functor from R-modules to types preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget (Module.{w} R)) | Module.forget_preserves_limits_of_size.{w w} | instance | Module.forget_preserves_limits | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
direct_limit_diagram : ι ⥤ Module R | { obj := λ i, Module.of R (G i),
map := λ i j hij, f i j hij.le,
map_id' := λ i, by { apply linear_map.ext, intro x, apply module.directed_system.map_self },
map_comp' := λ i j k hij hjk,
begin
apply linear_map.ext,
intro x,
symmetry,
apply module.directed_system.map_map
end } | def | Module.direct_limit_diagram | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module",
"Module.of",
"linear_map.ext",
"module.directed_system.map_map",
"module.directed_system.map_self"
] | The diagram (in the sense of `category_theory`)
of an unbundled `direct_limit` of modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit_cocone : cocone (direct_limit_diagram G f) | { X := Module.of R $ direct_limit G f,
ι := { app := module.direct_limit.of R ι G f,
naturality' := λ i j hij, by { apply linear_map.ext, intro x, exact direct_limit.of_f } } } | def | Module.direct_limit_cocone | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"Module.of",
"linear_map.ext",
"module.direct_limit.of"
] | The `cocone` on `direct_limit_diagram` corresponding to
the unbundled `direct_limit` of modules.
In `direct_limit_is_colimit` we show that it is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit_is_colimit [nonempty ι] [is_directed ι (≤)] :
is_colimit (direct_limit_cocone G f) | { desc := λ s, direct_limit.lift R ι G f s.ι.app $ λ i j h x, by { rw [←s.w (hom_of_le h)], refl },
fac' := λ s i,
begin
apply linear_map.ext,
intro x,
dsimp,
exact direct_limit.lift_of s.ι.app _ x,
end,
uniq' := λ s m h,
begin
have : s.ι.app = λ i, linear_map.comp m (direct_limit.of R ι (... | def | Module.direct_limit_is_colimit | algebra.category.Module | src/algebra/category/Module/limits.lean | [
"algebra.category.Module.basic",
"algebra.category.Group.limits",
"algebra.direct_limit"
] | [
"is_directed",
"linear_map.comp",
"linear_map.ext",
"module.direct_limit.lift_unique"
] | The unbundled `direct_limit` of modules is a colimit
in the sense of `category_theory`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_cone : fan Z | fan.mk (Module.of R (Π i : ι, Z i)) (λ i, (linear_map.proj i : (Π i : ι, Z i) →ₗ[R] Z i)) | def | Module.product_cone | algebra.category.Module | src/algebra/category/Module/products.lean | [
"linear_algebra.pi",
"algebra.category.Module.basic"
] | [
"Module.of",
"linear_map.proj"
] | The product cone induced by the concrete product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_cone_is_limit : is_limit (product_cone Z) | { lift := λ s, (linear_map.pi (λ j, s.π.app ⟨j⟩) : s.X →ₗ[R] (Π i : ι, Z i)),
fac' := λ s j, by { cases j, tidy, },
uniq' := λ s m w, by { ext x i, exact linear_map.congr_fun (w ⟨i⟩) x, }, } | def | Module.product_cone_is_limit | algebra.category.Module | src/algebra/category/Module/products.lean | [
"linear_algebra.pi",
"algebra.category.Module.basic"
] | [
"lift",
"linear_map.congr_fun",
"linear_map.pi"
] | The concrete product cone is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso_pi :
∏ Z ≅ Module.of R (Π i, Z i) | limit.iso_limit_cone ⟨_, product_cone_is_limit Z⟩ | def | Module.pi_iso_pi | algebra.category.Module | src/algebra/category/Module/products.lean | [
"linear_algebra.pi",
"algebra.category.Module.basic"
] | [
"Module.of"
] | The categorical product of a family of objects in `Module`
agrees with the usual module-theoretical product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso_pi_inv_kernel_ι (i : ι) :
(pi_iso_pi Z).inv ≫ pi.π Z i = (linear_map.proj i : (Π i : ι, Z i) →ₗ[R] Z i) | limit.iso_limit_cone_inv_π _ _ | lemma | Module.pi_iso_pi_inv_kernel_ι | algebra.category.Module | src/algebra/category/Module/products.lean | [
"linear_algebra.pi",
"algebra.category.Module.basic"
] | [
"linear_map.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_iso_pi_hom_ker_subtype (i : ι) :
(pi_iso_pi Z).hom ≫ (linear_map.proj i : (Π i : ι, Z i) →ₗ[R] Z i) = pi.π Z i | is_limit.cone_point_unique_up_to_iso_inv_comp _ (limit.is_limit _) (discrete.mk i) | lemma | Module.pi_iso_pi_hom_ker_subtype | algebra.category.Module | src/algebra/category/Module/products.lean | [
"linear_algebra.pi",
"algebra.category.Module.basic"
] | [
"linear_map.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_projective.iff_projective {R : Type u} [ring R]
{P : Type (max u v)} [add_comm_group P] [module R P] :
module.projective R P ↔ projective (Module.of R P) | begin
refine ⟨λ h, _, λ h, _⟩,
{ letI : module.projective R ↥(Module.of R P) := h,
exact ⟨λ E X f e epi, module.projective_lifting_property _ _
((Module.epi_iff_surjective _).mp epi)⟩ },
{ refine module.projective_of_lifting_property _,
introsI E X mE mX sE sX f g s,
haveI : epi ↟f := (Module.ep... | theorem | is_projective.iff_projective | algebra.category.Module | src/algebra/category/Module/projective.lean | [
"algebra.category.Module.epi_mono",
"algebra.module.projective",
"category_theory.preadditive.projective",
"linear_algebra.finsupp_vector_space"
] | [
"Module.epi_iff_surjective",
"Module.of",
"add_comm_group",
"module",
"module.projective",
"module.projective_lifting_property",
"module.projective_of_lifting_property",
"ring"
] | The categorical notion of projective object agrees with the explicit module-theoretic notion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_of_free {ι : Type*} (b : basis ι R M) : projective M | projective.of_iso (Module.of_self_iso _)
((is_projective.iff_projective).mp (module.projective_of_basis b)) | lemma | Module.projective_of_free | algebra.category.Module | src/algebra/category/Module/projective.lean | [
"algebra.category.Module.epi_mono",
"algebra.module.projective",
"category_theory.preadditive.projective",
"linear_algebra.finsupp_vector_space"
] | [
"Module.of_self_iso",
"basis",
"is_projective.iff_projective",
"module.projective_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Module_enough_projectives : enough_projectives (Module.{max u v} R) | { presentation :=
λ M,
⟨{ P := Module.of R (M →₀ R),
projective := projective_of_free finsupp.basis_single_one,
f := finsupp.basis_single_one.constr ℕ id,
epi := (epi_iff_range_eq_top _).mpr
(range_eq_top.2 (λ m, ⟨finsupp.single m (1 : R), by simp [basis.constr]⟩)) }⟩, } | instance | Module.Module_enough_projectives | algebra.category.Module | src/algebra/category/Module/projective.lean | [
"algebra.category.Module.epi_mono",
"algebra.module.projective",
"category_theory.preadditive.projective",
"linear_algebra.finsupp_vector_space"
] | [
"Module.of",
"basis.constr",
"finsupp.basis_single_one"
] | The category of modules has enough projectives, since every module is a quotient of a free
module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_iff_is_simple_module : simple (of R M) ↔ is_simple_module R M | (simple_iff_subobject_is_simple_order _).trans (subobject_Module (of R M)).is_simple_order_iff | lemma | simple_iff_is_simple_module | algebra.category.Module | src/algebra/category/Module/simple.lean | [
"category_theory.simple",
"algebra.category.Module.subobject",
"algebra.category.Module.algebra",
"ring_theory.simple_module",
"linear_algebra.finite_dimensional"
] | [
"is_simple_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple_iff_is_simple_module' (M : Module R) : simple M ↔ is_simple_module R M | (simple.iff_of_iso (of_self_iso M).symm).trans simple_iff_is_simple_module | lemma | simple_iff_is_simple_module' | algebra.category.Module | src/algebra/category/Module/simple.lean | [
"category_theory.simple",
"algebra.category.Module.subobject",
"algebra.category.Module.algebra",
"ring_theory.simple_module",
"linear_algebra.finite_dimensional"
] | [
"Module",
"is_simple_module",
"simple_iff_is_simple_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple_of_is_simple_module [is_simple_module R M] : simple (of R M) | simple_iff_is_simple_module.mpr ‹_› | instance | simple_of_is_simple_module | algebra.category.Module | src/algebra/category/Module/simple.lean | [
"category_theory.simple",
"algebra.category.Module.subobject",
"algebra.category.Module.algebra",
"ring_theory.simple_module",
"linear_algebra.finite_dimensional"
] | [
"is_simple_module"
] | A simple module is a simple object in the category of modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_simple_module_of_simple (M : Module R) [simple M] : is_simple_module R M | simple_iff_is_simple_module.mp (simple.of_iso (of_self_iso M)) | instance | is_simple_module_of_simple | algebra.category.Module | src/algebra/category/Module/simple.lean | [
"category_theory.simple",
"algebra.category.Module.subobject",
"algebra.category.Module.algebra",
"ring_theory.simple_module",
"linear_algebra.finite_dimensional"
] | [
"Module",
"is_simple_module"
] | A simple object in the category of modules is a simple module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_of_finrank_eq_one {k : Type*} [field k] [algebra k R]
{V : Module R} (h : finrank k V = 1) : simple V | (simple_iff_is_simple_module' V).mpr (is_simple_module_of_finrank_eq_one h) | lemma | simple_of_finrank_eq_one | algebra.category.Module | src/algebra/category/Module/simple.lean | [
"category_theory.simple",
"algebra.category.Module.subobject",
"algebra.category.Module.algebra",
"ring_theory.simple_module",
"linear_algebra.finite_dimensional"
] | [
"Module",
"algebra",
"field",
"is_simple_module_of_finrank_eq_one",
"simple_iff_is_simple_module'"
] | Any `k`-algebra module which is 1-dimensional over `k` is simple. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subobject_Module : subobject M ≃o submodule R M | order_iso.symm
({ inv_fun := λ S, S.arrow.range,
to_fun := λ N, subobject.mk ↾N.subtype,
right_inv := λ S, eq.symm
begin
fapply eq_mk_of_comm,
{ apply linear_equiv.to_Module_iso'_left,
apply linear_equiv.of_bijective (linear_map.cod_restrict S.arrow.range S.arrow _),
split,
{ simpa only ... | def | Module.subobject_Module | algebra.category.Module | src/algebra/category/Module/subobject.lean | [
"algebra.category.Module.epi_mono",
"algebra.category.Module.kernels",
"category_theory.subobject.well_powered",
"category_theory.subobject.limits"
] | [
"inv_fun",
"linear_equiv.of_bijective",
"linear_equiv.range_comp",
"linear_equiv.to_Module_iso'_left",
"linear_map.cod_restrict",
"linear_map.ext",
"linear_map.ker_cod_restrict",
"linear_map.ker_eq_bot",
"linear_map.mem_range_self",
"linear_map.range",
"linear_map.range_cod_restrict",
"linear_... | The categorical subobjects of a module `M` are in one-to-one correspondence with its
submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
well_powered_Module : well_powered (Module.{v} R) | ⟨λ M, ⟨⟨_, ⟨(subobject_Module M).to_equiv⟩⟩⟩⟩ | instance | Module.well_powered_Module | algebra.category.Module | src/algebra/category/Module/subobject.lean | [
"algebra.category.Module.epi_mono",
"algebra.category.Module.kernels",
"category_theory.subobject.well_powered",
"category_theory.subobject.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_kernel_subobject {M N : Module R} {f : M ⟶ N} :
linear_map.ker f →ₗ[R] kernel_subobject f | (kernel_subobject_iso f ≪≫ Module.kernel_iso_ker f).inv | def | Module.to_kernel_subobject | algebra.category.Module | src/algebra/category/Module/subobject.lean | [
"algebra.category.Module.epi_mono",
"algebra.category.Module.kernels",
"category_theory.subobject.well_powered",
"category_theory.subobject.limits"
] | [
"Module",
"Module.kernel_iso_ker",
"linear_map.ker"
] | Bundle an element `m : M` such that `f m = 0` as a term of `kernel_subobject f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_kernel_subobject_arrow {M N : Module R} {f : M ⟶ N} (x : linear_map.ker f) :
(kernel_subobject f).arrow (to_kernel_subobject x) = x.1 | by simp [to_kernel_subobject] | lemma | Module.to_kernel_subobject_arrow | algebra.category.Module | src/algebra/category/Module/subobject.lean | [
"algebra.category.Module.epi_mono",
"algebra.category.Module.kernels",
"category_theory.subobject.well_powered",
"category_theory.subobject.limits"
] | [
"Module",
"linear_map.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_π_image_subobject_ext {L M N : Module.{v} R}
(f : L ⟶ M) [has_image f] (g : (image_subobject f : Module.{v} R) ⟶ N) [has_cokernel g]
{x y : N} (l : L) (w : x = y + g (factor_thru_image_subobject f l)) :
cokernel.π g x = cokernel.π g y | by { subst w, simp, } | lemma | Module.cokernel_π_image_subobject_ext | algebra.category.Module | src/algebra/category/Module/subobject.lean | [
"algebra.category.Module.epi_mono",
"algebra.category.Module.kernels",
"category_theory.subobject.well_powered",
"category_theory.subobject.limits"
] | [] | An extensionality lemma showing that two elements of a cokernel by an image
are equal if they differ by an element of the image.
The application is for homology:
two elements in homology are equal if they differ by a boundary. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_equiv_End_forget₂ (R : Type u) [ring R] :
R ≃+* End (AdditiveFunctor.of (forget₂ (Module.{u} R) AddCommGroup.{u})) | { to_fun := λ r,
{ app := λ M, by apply distrib_mul_action.to_add_monoid_hom M r,
naturality' := λ M N f, by { ext, exact (f.map_smul _ _).symm, }, },
inv_fun := λ φ, φ.app (Module.of R R) (1 : R),
left_inv := by { intros r, simp, },
right_inv := begin
intros φ, ext M x,
simp only [distrib_mul_actio... | def | ring_equiv_End_forget₂ | algebra.category.Module | src/algebra/category/Module/tannaka.lean | [
"algebra.category.Module.basic",
"linear_algebra.span"
] | [
"Module.as_hom_right",
"Module.of",
"add_smul",
"distrib_mul_action.to_add_monoid_hom",
"inv_fun",
"linear_map.to_span_singleton",
"one_smul",
"ring"
] | An ingredient of Tannaka duality for rings:
A ring `R` is equivalent to
the endomorphisms of the additive forgetful functor `Module R ⥤ AddCommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tensor_obj (M N : Module R) : Module R | Module.of R (M ⊗[R] N) | def | Module.monoidal_category.tensor_obj | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"Module.of"
] | (implementation) tensor product of R-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tensor_hom {M N M' N' : Module R} (f : M ⟶ N) (g : M' ⟶ N') :
tensor_obj M M' ⟶ tensor_obj N N' | tensor_product.map f g | def | Module.monoidal_category.tensor_hom | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"tensor_product.map"
] | (implementation) tensor product of morphisms R-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tensor_id (M N : Module R) : tensor_hom (𝟙 M) (𝟙 N) = 𝟙 (Module.of R (M ⊗ N)) | by { ext1, refl } | lemma | Module.monoidal_category.tensor_id | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"Module.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Module R}
(f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂) :
tensor_hom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensor_hom f₁ f₂ ≫ tensor_hom g₁ g₂ | by { ext1, refl } | lemma | Module.monoidal_category.tensor_comp | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator (M : Module.{v} R) (N : Module.{w} R) (K : Module.{x} R) :
tensor_obj (tensor_obj M N) K ≅ tensor_obj M (tensor_obj N K) | (tensor_product.assoc R M N K).to_Module_iso | def | Module.monoidal_category.associator | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"tensor_product.assoc"
] | (implementation) the associator for R-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associator_naturality_aux
{X₁ X₂ X₃ : Type*}
[add_comm_monoid X₁] [add_comm_monoid X₂] [add_comm_monoid X₃]
[module R X₁] [module R X₂] [module R X₃]
{Y₁ Y₂ Y₃ : Type*}
[add_comm_monoid Y₁] [add_comm_monoid Y₂] [add_comm_monoid Y₃]
[module R Y₁] [module R Y₂] [module R Y₃]
(f₁ : X₁ →ₗ[R] Y₁) (f₂ : X₂ →ₗ[R... | begin
apply tensor_product.ext_threefold,
intros x y z,
refl
end | lemma | Module.monoidal_category.associator_naturality_aux | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"add_comm_monoid",
"module",
"tensor_product.ext_threefold"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_aux
(W X Y Z : Type*)
[add_comm_monoid W] [add_comm_monoid X] [add_comm_monoid Y] [add_comm_monoid Z]
[module R W] [module R X] [module R Y] [module R Z] :
((map (1 : W →ₗ[R] W) (assoc R X Y Z).to_linear_map).comp (assoc R W (X ⊗[R] Y) Z).to_linear_map)
.comp (map ↑(assoc R W X Y) (1 : Z →ₗ[R] Z)) ... | begin
apply tensor_product.ext_fourfold,
intros w x y z,
refl
end | lemma | Module.monoidal_category.pentagon_aux | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"add_comm_monoid",
"module",
"tensor_product.ext_fourfold"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : Module R}
(f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :
tensor_hom (tensor_hom f₁ f₂) f₃ ≫ (associator Y₁ Y₂ Y₃).hom =
(associator X₁ X₂ X₃).hom ≫ tensor_hom f₁ (tensor_hom f₂ f₃) | by convert associator_naturality_aux f₁ f₂ f₃ using 1 | lemma | Module.monoidal_category.associator_naturality | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon (W X Y Z : Module R) :
tensor_hom (associator W X Y).hom (𝟙 Z) ≫ (associator W (tensor_obj X Y) Z).hom
≫ tensor_hom (𝟙 W) (associator X Y Z).hom =
(associator (tensor_obj W X) Y Z).hom ≫ (associator W X (tensor_obj Y Z)).hom | by convert pentagon_aux R W X Y Z using 1 | lemma | Module.monoidal_category.pentagon | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor (M : Module.{u} R) : Module.of R (R ⊗[R] M) ≅ M | (linear_equiv.to_Module_iso (tensor_product.lid R M) : of R (R ⊗ M) ≅ of R M).trans (of_self_iso M) | def | Module.monoidal_category.left_unitor | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module.of",
"linear_equiv.to_Module_iso",
"tensor_product.lid"
] | (implementation) the left unitor for R-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_unitor_naturality {M N : Module R} (f : M ⟶ N) :
tensor_hom (𝟙 (Module.of R R)) f ≫ (left_unitor N).hom = (left_unitor M).hom ≫ f | begin
ext x y, dsimp,
erw [tensor_product.lid_tmul, tensor_product.lid_tmul],
rw linear_map.map_smul,
refl,
end | lemma | Module.monoidal_category.left_unitor_naturality | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"Module.of",
"linear_map.map_smul",
"tensor_product.lid_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor (M : Module.{u} R) : Module.of R (M ⊗[R] R) ≅ M | (linear_equiv.to_Module_iso (tensor_product.rid R M) : of R (M ⊗ R) ≅ of R M).trans (of_self_iso M) | def | Module.monoidal_category.right_unitor | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module.of",
"linear_equiv.to_Module_iso",
"tensor_product.rid"
] | (implementation) the right unitor for R-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_unitor_naturality {M N : Module R} (f : M ⟶ N) :
tensor_hom f (𝟙 (Module.of R R)) ≫ (right_unitor N).hom = (right_unitor M).hom ≫ f | begin
ext x y, dsimp,
erw [tensor_product.rid_tmul, tensor_product.rid_tmul],
rw linear_map.map_smul,
refl,
end | lemma | Module.monoidal_category.right_unitor_naturality | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"Module.of",
"linear_map.map_smul",
"tensor_product.rid_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triangle (M N : Module.{u} R) :
(associator M (Module.of R R) N).hom ≫ tensor_hom (𝟙 M) (left_unitor N).hom =
tensor_hom (right_unitor M).hom (𝟙 N) | begin
apply tensor_product.ext_threefold,
intros x y z,
change R at y,
dsimp [tensor_hom, associator],
erw [tensor_product.lid_tmul, tensor_product.rid_tmul],
exact (tensor_product.smul_tmul _ _ _).symm
end | lemma | Module.monoidal_category.triangle | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module.of",
"tensor_product.ext_threefold",
"tensor_product.lid_tmul",
"tensor_product.rid_tmul",
"tensor_product.smul_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_category : monoidal_category (Module.{u} R) | { -- data
tensor_obj := tensor_obj,
tensor_hom := @tensor_hom _ _,
tensor_unit := Module.of R R,
associator := associator,
left_unitor := left_unitor,
right_unitor := right_unitor,
-- properties
tensor_id' := λ M N, tensor_id M N,
tensor_comp' := λ M N K M' N' K' f g ... | instance | Module.monoidal_category | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_apply {K L M N : Module.{u} R} (f : K ⟶ L) (g : M ⟶ N) (k : K) (m : M) :
(f ⊗ g) (k ⊗ₜ m) = f k ⊗ₜ g m | rfl | lemma | Module.monoidal_category.hom_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_hom_apply {M : Module.{u} R} (r : R) (m : M) :
((λ_ M).hom : 𝟙_ (Module R) ⊗ M ⟶ M) (r ⊗ₜ[R] m) = r • m | tensor_product.lid_tmul m r | lemma | Module.monoidal_category.left_unitor_hom_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"tensor_product.lid_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_inv_apply {M : Module.{u} R} (m : M) :
((λ_ M).inv : M ⟶ 𝟙_ (Module.{u} R) ⊗ M) m = 1 ⊗ₜ[R] m | tensor_product.lid_symm_apply m | lemma | Module.monoidal_category.left_unitor_inv_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"tensor_product.lid_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_hom_apply {M : Module.{u} R} (m : M) (r : R) :
((ρ_ M).hom : M ⊗ 𝟙_ (Module R) ⟶ M) (m ⊗ₜ r) = r • m | tensor_product.rid_tmul m r | lemma | Module.monoidal_category.right_unitor_hom_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"Module",
"tensor_product.rid_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_inv_apply {M : Module.{u} R} (m : M) :
((ρ_ M).inv : M ⟶ M ⊗ 𝟙_ (Module.{u} R)) m = m ⊗ₜ[R] 1 | tensor_product.rid_symm_apply m | lemma | Module.monoidal_category.right_unitor_inv_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [
"tensor_product.rid_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_hom_apply {M N K : Module.{u} R} (m : M) (n : N) (k : K) :
((α_ M N K).hom : (M ⊗ N) ⊗ K ⟶ M ⊗ (N ⊗ K)) ((m ⊗ₜ n) ⊗ₜ k) = (m ⊗ₜ (n ⊗ₜ k)) | rfl | lemma | Module.monoidal_category.associator_hom_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_inv_apply {M N K : Module.{u} R} (m : M) (n : N) (k : K) :
((α_ M N K).inv : M ⊗ (N ⊗ K) ⟶ (M ⊗ N) ⊗ K) (m ⊗ₜ (n ⊗ₜ k)) = ((m ⊗ₜ n) ⊗ₜ k) | rfl | lemma | Module.monoidal_category.associator_inv_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/basic.lean | [
"algebra.category.Module.basic",
"linear_algebra.tensor_product",
"category_theory.linear.yoneda",
"category_theory.monoidal.linear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed_hom_equiv (M N P : Module.{u} R) :
((monoidal_category.tensor_left M).obj N ⟶ P) ≃
(N ⟶ ((linear_coyoneda R (Module R)).obj (op M)).obj P) | { to_fun := λ f, linear_map.compr₂ (tensor_product.mk R N M) ((β_ N M).hom ≫ f),
inv_fun := λ f, (β_ M N).hom ≫ tensor_product.lift f,
left_inv := λ f, begin ext m n,
simp only [tensor_product.mk_apply, tensor_product.lift.tmul, linear_map.compr₂_apply,
function.comp_app, coe_comp, monoidal_category.braid... | def | Module.monoidal_closed_hom_equiv | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [
"Module",
"inv_fun",
"linear_map.compr₂",
"linear_map.compr₂_apply",
"tensor_product.lift",
"tensor_product.lift.tmul",
"tensor_product.mk",
"tensor_product.mk_apply"
] | Auxiliary definition for the `monoidal_closed` instance on `Module R`.
(This is only a separate definition in order to speed up typechecking. ) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom_map_apply {M N P : Module.{u} R} (f : N ⟶ P) (g : Module.of R (M ⟶ N)) :
(ihom M).map f g = g ≫ f | rfl | lemma | Module.ihom_map_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [
"Module.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed_curry {M N P : Module.{u} R} (f : M ⊗ N ⟶ P) (x : M) (y : N) :
@coe_fn _ _ linear_map.has_coe_to_fun ((monoidal_closed.curry f : N →ₗ[R] (M →ₗ[R] P)) y) x =
f (x ⊗ₜ[R] y) | rfl | lemma | Module.monoidal_closed_curry | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed_uncurry {M N P : Module.{u} R}
(f : N ⟶ (M ⟶[Module.{u} R] P)) (x : M) (y : N) :
monoidal_closed.uncurry f (x ⊗ₜ[R] y) = (@coe_fn _ _ linear_map.has_coe_to_fun (f y)) x | rfl | lemma | Module.monoidal_closed_uncurry | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_ev_app (M N : Module.{u} R) :
(ihom.ev M).app N = tensor_product.uncurry _ _ _ _ linear_map.id.flip | begin
ext,
exact Module.monoidal_closed_uncurry _ _ _,
end | lemma | Module.ihom_ev_app | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [
"Module.monoidal_closed_uncurry",
"tensor_product.uncurry"
] | Describes the counit of the adjunction `M ⊗ - ⊣ Hom(M, -)`. Given an `R`-module `N` this
should give a map `M ⊗ Hom(M, N) ⟶ N`, so we flip the order of the arguments in the identity map
`Hom(M, N) ⟶ (M ⟶ N)` and uncurry the resulting map `M ⟶ Hom(M, N) ⟶ N.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom_coev_app (M N : Module.{u} R) :
(ihom.coev M).app N = (tensor_product.mk _ _ _).flip | rfl | lemma | Module.ihom_coev_app | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [
"tensor_product.mk"
] | Describes the unit of the adjunction `M ⊗ - ⊣ Hom(M, -)`. Given an `R`-module `N` this should
define a map `N ⟶ Hom(M, M ⊗ N)`, which is given by flipping the arguments in the natural
`R`-bilinear map `M ⟶ N ⟶ M ⊗ N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoidal_closed_pre_app {M N : Module.{u} R} (P : Module.{u} R) (f : N ⟶ M) :
(monoidal_closed.pre f).app P = linear_map.lcomp R _ f | rfl | lemma | Module.monoidal_closed_pre_app | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/closed.lean | [
"category_theory.closed.monoidal",
"algebra.category.Module.monoidal.symmetric"
] | [
"linear_map.lcomp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
braiding (M N : Module.{u} R) : (M ⊗ N) ≅ (N ⊗ M) | linear_equiv.to_Module_iso (tensor_product.comm R M N) | def | Module.braiding | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [
"linear_equiv.to_Module_iso",
"tensor_product.comm"
] | (implementation) the braiding for R-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
braiding_naturality {X₁ X₂ Y₁ Y₂ : Module.{u} R} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) :
(f ⊗ g) ≫ (Y₁.braiding Y₂).hom =
(X₁.braiding X₂).hom ≫ (g ⊗ f) | begin
apply tensor_product.ext',
intros x y,
refl
end | lemma | Module.monoidal_category.braiding_naturality | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [
"tensor_product.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hexagon_forward (X Y Z : Module.{u} R) :
(α_ X Y Z).hom ≫ (braiding X _).hom ≫ (α_ Y Z X).hom =
((braiding X Y).hom ⊗ 𝟙 Z) ≫ (α_ Y X Z).hom ≫ (𝟙 Y ⊗ (braiding X Z).hom) | begin
apply tensor_product.ext_threefold,
intros x y z,
refl,
end | lemma | Module.monoidal_category.hexagon_forward | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [
"tensor_product.ext_threefold"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hexagon_reverse (X Y Z : Module.{u} R) :
(α_ X Y Z).inv ≫ (braiding _ Z).hom ≫ (α_ Z X Y).inv =
(𝟙 X ⊗ (Y.braiding Z).hom) ≫ (α_ X Z Y).inv ≫ ((X.braiding Z).hom ⊗ 𝟙 Y) | begin
apply (cancel_epi (α_ X Y Z).hom).1,
apply tensor_product.ext_threefold,
intros x y z,
refl,
end | lemma | Module.monoidal_category.hexagon_reverse | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [
"tensor_product.ext_threefold"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetric_category : symmetric_category (Module.{u} R) | { braiding := braiding,
braiding_naturality' := λ X₁ X₂ Y₁ Y₂ f g, braiding_naturality f g,
hexagon_forward' := hexagon_forward,
hexagon_reverse' := hexagon_reverse, } | instance | Module.monoidal_category.symmetric_category | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [] | The symmetric monoidal structure on `Module R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
braiding_hom_apply {M N : Module.{u} R} (m : M) (n : N) :
((β_ M N).hom : M ⊗ N ⟶ N ⊗ M) (m ⊗ₜ n) = n ⊗ₜ m | rfl | lemma | Module.monoidal_category.braiding_hom_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
braiding_inv_apply {M N : Module.{u} R} (m : M) (n : N) :
((β_ M N).inv : N ⊗ M ⟶ M ⊗ N) (n ⊗ₜ m) = m ⊗ₜ n | rfl | lemma | Module.monoidal_category.braiding_inv_apply | algebra.category.Module.monoidal | src/algebra/category/Module/monoidal/symmetric.lean | [
"category_theory.monoidal.braided",
"algebra.category.Module.monoidal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_one : Semigroup.{u} ⥤ Mon.{u} | { obj := λ S, Mon.of (with_one S),
map := λ X Y, with_one.map,
map_id' := λ X, with_one.map_id,
map_comp' := λ X Y Z, with_one.map_comp } | def | adjoin_one | algebra.category.Mon | src/algebra/category/Mon/adjunctions.lean | [
"algebra.category.Mon.basic",
"algebra.category.Semigroup.basic",
"algebra.group.with_one.basic",
"algebra.free_monoid.basic"
] | [
"Mon.of",
"with_one",
"with_one.map",
"with_one.map_comp",
"with_one.map_id"
] | The functor of adjoining a neutral element `one` to a semigroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_forget_to_Semigroup : has_forget₂ Mon Semigroup | { forget₂ :=
{ obj := λ M, Semigroup.of M,
map := λ M N, monoid_hom.to_mul_hom }, } | instance | has_forget_to_Semigroup | algebra.category.Mon | src/algebra/category/Mon/adjunctions.lean | [
"algebra.category.Mon.basic",
"algebra.category.Semigroup.basic",
"algebra.group.with_one.basic",
"algebra.free_monoid.basic"
] | [
"Mon",
"Semigroup",
"Semigroup.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_one_adj : adjoin_one ⊣ forget₂ Mon.{u} Semigroup.{u} | adjunction.mk_of_hom_equiv
{ hom_equiv := λ S M, with_one.lift.symm,
hom_equiv_naturality_left_symm' :=
begin
intros S T M f g,
ext,
simp only [equiv.symm_symm, adjoin_one_map, coe_comp],
simp_rw with_one.map,
apply with_one.cases_on x,
{ refl },
{ simp }
end } | def | adjoin_one_adj | algebra.category.Mon | src/algebra/category/Mon/adjunctions.lean | [
"algebra.category.Mon.basic",
"algebra.category.Semigroup.basic",
"algebra.group.with_one.basic",
"algebra.free_monoid.basic"
] | [
"adjoin_one",
"equiv.symm_symm",
"with_one.cases_on",
"with_one.map"
] | The adjoin_one-forgetful adjunction from `Semigroup` to `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free : Type u ⥤ Mon.{u} | { obj := λ α, Mon.of (free_monoid α),
map := λ X Y, free_monoid.map,
map_id' := by { intros, ext1, refl },
map_comp' := by { intros, ext1, refl } } | def | free | algebra.category.Mon | src/algebra/category/Mon/adjunctions.lean | [
"algebra.category.Mon.basic",
"algebra.category.Semigroup.basic",
"algebra.group.with_one.basic",
"algebra.free_monoid.basic"
] | [
"Mon.of",
"free_monoid",
"free_monoid.map"
] | The free functor `Type u ⥤ Mon` sending a type `X` to the free monoid on `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj : free ⊣ forget Mon.{u} | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X G, free_monoid.lift.symm,
hom_equiv_naturality_left_symm' := λ X Y G f g, by { ext1, refl } } | def | adj | algebra.category.Mon | src/algebra/category/Mon/adjunctions.lean | [
"algebra.category.Mon.basic",
"algebra.category.Semigroup.basic",
"algebra.group.with_one.basic",
"algebra.free_monoid.basic"
] | [
"free"
] | The free-forgetful adjunction for monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Mon : Type (u+1) | bundled monoid | def | Mon | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"monoid"
] | The category of monoids and monoid morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
assoc_monoid_hom (M N : Type*) [monoid M] [monoid N] | monoid_hom M N | abbreviation | Mon.assoc_monoid_hom | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"monoid",
"monoid_hom"
] | `monoid_hom` doesn't actually assume associativity. This alias is needed to make the category
theory machinery work. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom : bundled_hom assoc_monoid_hom | ⟨λ M N [monoid M] [monoid N], by exactI @monoid_hom.to_fun M N _ _,
λ M [monoid M], by exactI @monoid_hom.id M _,
λ M N P [monoid M] [monoid N] [monoid P], by exactI @monoid_hom.comp M N P _ _ _,
λ M N [monoid M] [monoid N], by exactI @monoid_hom.coe_inj M N _ _⟩ | instance | Mon.bundled_hom | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"monoid",
"monoid_hom.coe_inj",
"monoid_hom.comp",
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (M : Type u) [monoid M] : Mon | bundled.of M | def | Mon.of | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"Mon",
"monoid"
] | Construct a bundled `Mon` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {X Y : Type u} [monoid X] [monoid Y] (f : X →* Y) :
of X ⟶ of Y | f | def | Mon.of_hom | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"monoid"
] | Typecheck a `monoid_hom` as a morphism in `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {X Y : Type u} [monoid X] [monoid Y] (f : X →* Y)
(x : X) : of_hom f x = f x | rfl | lemma | Mon.of_hom_apply | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [monoid R] : (Mon.of R : Type u) = R | rfl | lemma | Mon.coe_of | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"Mon.of",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommMon : Type (u+1) | bundled comm_monoid | def | CommMon | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"comm_monoid"
] | The category of commutative monoids and monoid morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (M : Type u) [comm_monoid M] : CommMon | bundled.of M | def | CommMon.of | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon",
"comm_monoid"
] | Construct a bundled `CommMon` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (R : Type u) [comm_monoid R] : (CommMon.of R : Type u) = R | rfl | lemma | CommMon.coe_of | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon.of",
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Mon : has_forget₂ CommMon Mon | bundled_hom.forget₂ _ _ | instance | CommMon.has_forget_to_Mon | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon",
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.to_Mon_iso (e : X ≃* Y) : Mon.of X ≅ Mon.of Y | { hom := e.to_monoid_hom,
inv := e.symm.to_monoid_hom } | def | mul_equiv.to_Mon_iso | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"Mon.of"
] | Build an isomorphism in the category `Mon` from a `mul_equiv` between `monoid`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.to_CommMon_iso (e : X ≃* Y) : CommMon.of X ≅ CommMon.of Y | { hom := e.to_monoid_hom,
inv := e.symm.to_monoid_hom } | def | mul_equiv.to_CommMon_iso | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon.of"
] | Build an isomorphism in the category `CommMon` from a `mul_equiv` between `comm_monoid`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Mon_iso_to_mul_equiv {X Y : Mon} (i : X ≅ Y) : X ≃* Y | i.hom.to_mul_equiv i.inv i.hom_inv_id i.inv_hom_id | def | category_theory.iso.Mon_iso_to_mul_equiv | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"Mon"
] | Build a `mul_equiv` from an isomorphism in the category `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommMon_iso_to_mul_equiv {X Y : CommMon} (i : X ≅ Y) : X ≃* Y | i.hom.to_mul_equiv i.inv i.hom_inv_id i.inv_hom_id | def | category_theory.iso.CommMon_iso_to_mul_equiv | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon"
] | Build a `mul_equiv` from an isomorphism in the category `CommMon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_iso_Mon_iso {X Y : Type u} [monoid X] [monoid Y] :
(X ≃* Y) ≅ (Mon.of X ≅ Mon.of Y) | { hom := λ e, e.to_Mon_iso,
inv := λ i, i.Mon_iso_to_mul_equiv, } | def | mul_equiv_iso_Mon_iso | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"Mon.of",
"monoid"
] | multiplicative equivalences between `monoid`s are the same as (isomorphic to) isomorphisms
in `Mon` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_iso_CommMon_iso {X Y : Type u} [comm_monoid X] [comm_monoid Y] :
(X ≃* Y) ≅ (CommMon.of X ≅ CommMon.of Y) | { hom := λ e, e.to_CommMon_iso,
inv := λ i, i.CommMon_iso_to_mul_equiv, } | def | mul_equiv_iso_CommMon_iso | algebra.category.Mon | src/algebra/category/Mon/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"algebra.punit_instances",
"category_theory.functor.reflects_isomorphisms"
] | [
"CommMon.of",
"comm_monoid"
] | multiplicative equivalences between `comm_monoid`s are the same as (isomorphic to) isomorphisms
in `CommMon` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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