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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biprod_iso_prod_inv_comp_snd (M N : Module.{v} R) :
(biprod_iso_prod M N).inv ≫ biprod.snd = linear_map.snd R M N | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.right) | lemma | Module.biprod_iso_prod_inv_comp_snd | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"linear_map.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (s : fan f) :
s.X ⟶ Module.of R (Π j, f j) | { to_fun := λ x j, s.π.app ⟨j⟩ x,
map_add' := λ x y, by { ext, simp, },
map_smul' := λ r x, by { ext, simp, }, } | def | Module.has_limit.lift | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"Module.of",
"lift"
] | The map from an arbitrary cone over a indexed family of abelian groups
to the cartesian product of those groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_limit_cone : limits.limit_cone (discrete.functor f) | { cone :=
{ X := Module.of R (Π j, f j),
π := discrete.nat_trans (λ j, (linear_map.proj j.as : (Π j, f j) →ₗ[R] f j.as)), },
is_limit :=
{ lift := lift f,
fac' := λ s j, by { cases j, ext, simp, },
uniq' := λ s m w,
begin
ext x j,
dsimp only [has_limit.lift],
simp only [linear_ma... | def | Module.has_limit.product_limit_cone | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"Module.of",
"lift",
"linear_map.coe_mk",
"linear_map.proj"
] | Construct limit data for a product in `Module R`, using `Module.of R (Π j, F.obj j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_iso_pi [fintype J] (f : J → Module.{v} R) :
(⨁ f : Module.{v} R) ≅ Module.of R (Π j, f j) | is_limit.cone_point_unique_up_to_iso
(biproduct.is_limit f)
(product_limit_cone f).is_limit | def | Module.biproduct_iso_pi | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"Module.of",
"fintype"
] | We verify that the biproduct we've just defined is isomorphic to the `Module R` structure
on the dependent function type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_iso_pi_inv_comp_π [fintype J]
(f : J → Module.{v} R) (j : J) :
(biproduct_iso_pi f).inv ≫ biproduct.π f j = (linear_map.proj j : (Π j, f j) →ₗ[R] f j) | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk j) | lemma | Module.biproduct_iso_pi_inv_comp_π | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"fintype",
"linear_map.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lequiv_prod_of_right_split_exact {f : B →ₗ[R] M}
(hj : function.injective j) (exac : j.range = g.ker) (h : g.comp f = linear_map.id) :
(A × B) ≃ₗ[R] M | (({ right_split := ⟨as_hom f, h⟩,
mono := (Module.mono_iff_injective $ as_hom j).mpr hj,
exact := (exact_iff _ _).mpr exac } : right_split _ _).splitting.iso.trans $
biprod_iso_prod _ _).to_linear_equiv.symm | def | lequiv_prod_of_right_split_exact | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"Module.mono_iff_injective",
"linear_map.id"
] | The isomorphism `A × B ≃ₗ[R] M` coming from a right split exact sequence `0 ⟶ A ⟶ M ⟶ B ⟶ 0`
of modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lequiv_prod_of_left_split_exact {f : M →ₗ[R] A}
(hg : function.surjective g) (exac : j.range = g.ker) (h : f.comp j = linear_map.id) :
(A × B) ≃ₗ[R] M | (({ left_split := ⟨as_hom f, h⟩,
epi := (Module.epi_iff_surjective $ as_hom g).mpr hg,
exact := (exact_iff _ _).mpr exac } : left_split _ _).splitting.iso.trans $
biprod_iso_prod _ _).to_linear_equiv.symm | def | lequiv_prod_of_left_split_exact | algebra.category.Module | src/algebra/category/Module/biproducts.lean | [
"algebra.group.pi",
"category_theory.limits.shapes.biproducts",
"algebra.category.Module.abelian",
"algebra.homology.short_exact.abelian"
] | [
"Module.epi_iff_surjective",
"linear_map.id"
] | The isomorphism `A × B ≃ₗ[R] M` coming from a left split exact sequence `0 ⟶ A ⟶ M ⟶ B ⟶ 0`
of modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj' : Module R | { carrier := M,
is_module := module.comp_hom M f } | def | category_theory.Module.restrict_scalars.obj' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"module.comp_hom"
] | Any `S`-module M is also an `R`-module via a ring homomorphism `f : R ⟶ S` by defining
`r • m := f r • m` (`module.comp_hom`). This is called restriction of scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map' {M M' : Module.{v} S} (g : M ⟶ M') :
obj' f M ⟶ obj' f M' | { map_smul' := λ r, g.map_smul (f r), ..g } | def | category_theory.Module.restrict_scalars.map' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [] | Given an `S`-linear map `g : M → M'` between `S`-modules, `g` is also `R`-linear between `M` and
`M'` by means of restriction of scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) :
Module.{v} S ⥤ Module.{v} R | { obj := restrict_scalars.obj' f,
map := λ _ _, restrict_scalars.map' f,
map_id' := λ _, linear_map.ext $ λ m, rfl,
map_comp' := λ _ _ _ g h, linear_map.ext $ λ m, rfl } | def | category_theory.Module.restrict_scalars | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"linear_map.ext",
"restrict_scalars",
"ring"
] | The restriction of scalars operation is functorial. For any `f : R →+* S` a ring homomorphism,
* an `S`-module `M` can be considered as `R`-module by `r • m = f r • m`
* an `S`-linear map is also `R`-linear | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.map_apply {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S)
{M M' : Module.{v} S} (g : M ⟶ M') (x) : (restrict_scalars f).map g x = g x | rfl | lemma | category_theory.Module.restrict_scalars.map_apply | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"restrict_scalars",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.smul_def {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S)
{M : Module.{v} S} (r : R) (m : (restrict_scalars f).obj M) : r • m = (f r • m : M) | rfl | lemma | category_theory.Module.restrict_scalars.smul_def | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"restrict_scalars",
"restrict_scalars.smul_def",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.smul_def' {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S)
{M : Module.{v} S} (r : R) (m : M) : (r • m : (restrict_scalars f).obj M) = (f r • m : M) | rfl | lemma | category_theory.Module.restrict_scalars.smul_def' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"restrict_scalars",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_mk {R : Type u₁} {S : Type u₂} [ring R] [comm_ring S] (f : R →+* S)
(M : Type v) [add_comm_group M] [module S M] :
@smul_comm_class R S M ((restrict_scalars.obj' f (Module.mk M)).is_module.to_has_smul) _ | { smul_comm := λ r s m, (by simp [←mul_smul, mul_comm] : f r • s • m = s • f r • m) } | instance | category_theory.Module.smul_comm_class_mk | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"add_comm_group",
"comm_ring",
"module",
"mul_comm",
"ring",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj' : Module S | ⟨tensor_product R ((restrict_scalars f).obj ⟨S⟩) M⟩ | def | category_theory.Module.extend_scalars.obj' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"restrict_scalars"
] | Extension of scalars turn an `R`-module into `S`-module by M ↦ S ⨂ M | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map' {M1 M2 : Module.{v} R} (l : M1 ⟶ M2) : (obj' f M1) ⟶ (obj' f M2) | -- The "by apply" part makes this require 75% fewer heartbeats to process (#16371).
by apply (@linear_map.base_change R S M1 M2 _ _ ((algebra_map S _).comp f).to_algebra _ _ _ _ l) | def | category_theory.Module.extend_scalars.map' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"algebra_map",
"linear_map.base_change"
] | Extension of scalars is a functor where an `R`-module `M` is sent to `S ⊗ M` and
`l : M1 ⟶ M2` is sent to `s ⊗ m ↦ s ⊗ l m` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map'_id {M : Module.{v} R} : map' f (𝟙 M) = 𝟙 _ | linear_map.ext $ λ (x : obj' f M),
begin
dsimp only [map', Module.id_apply],
induction x using tensor_product.induction_on with _ _ m s ihx ihy,
{ simp only [map_zero], },
{ rw [linear_map.base_change_tmul, Module.id_apply], },
{ rw [map_add, ihx, ihy] },
end | lemma | category_theory.Module.extend_scalars.map'_id | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module.id_apply",
"linear_map.base_change_tmul",
"linear_map.ext",
"tensor_product.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map'_comp {M₁ M₂ M₃ : Module.{v} R} (l₁₂ : M₁ ⟶ M₂) (l₂₃ : M₂ ⟶ M₃) :
map' f (l₁₂ ≫ l₂₃) = map' f l₁₂ ≫ map' f l₂₃ | linear_map.ext $ λ (x : obj' f M₁),
begin
dsimp only [map'],
induction x using tensor_product.induction_on with _ _ x y ihx ihy,
{ refl, },
{ refl, },
{ simp only [map_add, ihx, ihy], },
end | lemma | category_theory.Module.extend_scalars.map'_comp | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"linear_map.ext",
"tensor_product.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_scalars {R : Type u₁} {S : Type u₂} [comm_ring R] [comm_ring S] (f : R →+* S) :
Module.{v} R ⥤ Module.{max v u₂} S | { obj := λ M, extend_scalars.obj' f M,
map := λ M1 M2 l, extend_scalars.map' f l,
map_id' := λ _, extend_scalars.map'_id f,
map_comp' := λ _ _ _, extend_scalars.map'_comp f } | def | category_theory.Module.extend_scalars | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"comm_ring"
] | Extension of scalars is a functor where an `R`-module `M` is sent to `S ⊗ M` and
`l : M1 ⟶ M2` is sent to `s ⊗ m ↦ s ⊗ l m` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_tmul {M : Module.{v} R} (s s' : S) (m : M) :
s • (s' ⊗ₜ[R, f] m : (extend_scalars f).obj M) = (s * s') ⊗ₜ[R, f] m | rfl | lemma | category_theory.Module.extend_scalars.smul_tmul | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_tmul {M M' : Module.{v} R} (g : M ⟶ M') (s : S) (m : M) :
(extend_scalars f).map g (s ⊗ₜ[R, f] m) = s ⊗ₜ[R, f] g m | rfl | lemma | category_theory.Module.extend_scalars.map_tmul | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul : has_smul S $ S' →ₗ[R] M | { smul := λ s g,
{ to_fun := λ (s' : S), g (s' * s : S),
map_add' := λ (x y : S), by simp [add_mul, map_add],
map_smul' := λ r (t : S), by rw [ring_hom.id_apply, @restrict_scalars.smul_def _ _ _ _ f ⟨S⟩,
←linear_map.map_smul, @restrict_scalars.smul_def _ _ _ _ f ⟨S⟩, smul_eq_mul, smul_eq_mul,
mul_... | instance | category_theory.Module.coextend_scalars.has_smul | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"has_smul",
"mul_assoc",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_eq_mul"
] | Given an `R`-module M, consider Hom(S, M) -- the `R`-linear maps between S (as an `R`-module by
means of restriction of scalars) and M. `S` acts on Hom(S, M) by `s • g = x ↦ g (x • s)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_apply' (s : S) (g : S' →ₗ[R] M) (s' : S) :
@has_smul.smul _ _ (coextend_scalars.has_smul f _) s g s' = g (s' * s : S) | rfl | lemma | category_theory.Module.coextend_scalars.smul_apply' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action : mul_action S $ S' →ₗ[R] M | { one_smul := λ g, linear_map.ext $ λ (s : S), by simp,
mul_smul := λ (s t : S) g, linear_map.ext $ λ (x : S), by simp [mul_assoc],
..coextend_scalars.has_smul f _ } | instance | category_theory.Module.coextend_scalars.mul_action | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"linear_map.ext",
"mul_action",
"mul_assoc",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_mul_action : distrib_mul_action S $ S' →ₗ[R] M | { smul_add := λ s g h, linear_map.ext $ λ (t : S), by simp,
smul_zero := λ s, linear_map.ext $ λ (t : S), by simp,
..coextend_scalars.mul_action f _ } | instance | category_theory.Module.coextend_scalars.distrib_mul_action | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"distrib_mul_action",
"linear_map.ext",
"smul_add",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_module : module S $ S' →ₗ[R] M | { add_smul := λ s1 s2 g, linear_map.ext $ λ (x : S), by simp [mul_add],
zero_smul := λ g, linear_map.ext $ λ (x : S), by simp,
..coextend_scalars.distrib_mul_action f _ } | instance | category_theory.Module.coextend_scalars.is_module | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"add_smul",
"linear_map.ext",
"module",
"zero_smul"
] | `S` acts on Hom(S, M) by `s • g = x ↦ g (x • s)`, this action defines an `S`-module structure on
Hom(S, M). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj' : Module S | ⟨(restrict_scalars f).obj ⟨S⟩ →ₗ[R] M⟩ | def | category_theory.Module.coextend_scalars.obj' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"restrict_scalars"
] | If `M` is an `R`-module, then the set of `R`-linear maps `S →ₗ[R] M` is an `S`-module with
scalar multiplication defined by `s • l := x ↦ l (x • s)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map' {M M' : Module R} (g : M ⟶ M') : obj' f M ⟶ obj' f M' | { to_fun := λ h, g.comp h,
map_add' := λ _ _, linear_map.comp_add _ _ _,
map_smul' := λ s h, linear_map.ext $ λ (t : S), by simpa only [smul_apply'] } | def | category_theory.Module.coextend_scalars.map' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"linear_map.comp_add",
"linear_map.ext"
] | If `M, M'` are `R`-modules, then any `R`-linear map `g : M ⟶ M'` induces an `S`-linear map
`(S →ₗ[R] M) ⟶ (S →ₗ[R] M')` defined by `h ↦ g ∘ h` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coextend_scalars {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) :
Module R ⥤ Module S | { obj := coextend_scalars.obj' f,
map := λ _ _, coextend_scalars.map' f,
map_id' := λ M, linear_map.ext $ λ h, linear_map.ext $ λ x, rfl,
map_comp' := λ _ _ _ g h, linear_map.ext $ λ h, linear_map.ext $ λ x, rfl } | def | category_theory.Module.coextend_scalars | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"linear_map.ext",
"ring"
] | For any rings `R, S` and a ring homomorphism `f : R →+* S`, there is a functor from `R`-module to
`S`-module defined by `M ↦ (S →ₗ[R] M)` where `S` is considered as an `R`-module via restriction of
scalars and `g : M ⟶ M'` is sent to `h ↦ g ∘ h`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_apply (M : Module R) (g : (coextend_scalars f).obj M) (s s' : S) :
(s • g) s' = g (s' * s) | rfl | lemma | category_theory.Module.coextend_scalars.smul_apply | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_apply {M M' : Module R} (g : M ⟶ M') (x) (s : S) :
(coextend_scalars f).map g x s = g (x s) | rfl | lemma | category_theory.Module.coextend_scalars.map_apply | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv.from_restriction {X Y} (g : (restrict_scalars f).obj Y ⟶ X) :
Y ⟶ (coextend_scalars f).obj X | { to_fun := λ (y : Y),
{ to_fun := λ (s : S), g $ (s • y : Y),
map_add' := λ (s1 s2 : S), by simp [add_smul],
map_smul' := λ r (s : S), by rw [ring_hom.id_apply, ←g.map_smul,
@restrict_scalars.smul_def _ _ _ _ f ⟨S⟩, smul_eq_mul, mul_smul,
@restrict_scalars.smul_def _ _ _ _ f Y] },
map_add' := λ... | def | category_theory.Module.restriction_coextension_adj.hom_equiv.from_restriction | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"add_smul",
"linear_map.add_apply",
"linear_map.coe_mk",
"linear_map.ext",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_add",
"smul_eq_mul"
] | Given `R`-module X and `S`-module Y, any `g : (restrict_of_scalars f).obj Y ⟶ X`
corresponds to `Y ⟶ (coextend_scalars f).obj X` by sending `y ↦ (s ↦ g (s • y))` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv.to_restriction {X Y} (g : Y ⟶ (coextend_scalars f).obj X) :
(restrict_scalars f).obj Y ⟶ X | { to_fun := λ (y : Y), (g y).to_fun (1 : S),
map_add' := λ x y, by simp only [g.map_add, linear_map.to_fun_eq_coe, linear_map.add_apply],
map_smul' := λ r (y : Y), by rw [linear_map.to_fun_eq_coe, linear_map.to_fun_eq_coe,
ring_hom.id_apply, ←linear_map.map_smul, restrict_scalars.smul_def f r y,
@restrict_s... | def | category_theory.Module.restriction_coextension_adj.hom_equiv.to_restriction | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"linear_map.add_apply",
"linear_map.map_smul",
"linear_map.to_fun_eq_coe",
"mul_one",
"one_mul",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_eq_mul"
] | Given `R`-module X and `S`-module Y, any `g : Y ⟶ (coextend_scalars f).obj X`
corresponds to `(restrict_scalars f).obj Y ⟶ X` by `y ↦ g y 1` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit' : 𝟭 (Module S) ⟶ restrict_scalars f ⋙ coextend_scalars f | { app := λ Y,
{ to_fun := λ (y : Y),
{ to_fun := λ (s : S), (s • y : Y),
map_add' := λ s s', add_smul _ _ _,
map_smul' := λ r (s : S), by rw [ring_hom.id_apply, @restrict_scalars.smul_def _ _ _ _ f ⟨S⟩,
smul_eq_mul, mul_smul, restrict_scalars.smul_def f] },
map_add' := λ y1 y2, linear_map.... | def | category_theory.Module.restriction_coextension_adj.unit' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"add_smul",
"linear_map.add_apply",
"linear_map.coe_mk",
"linear_map.ext",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_add",
"smul_eq_mul"
] | The natural transformation from identity functor to the composition of restriction and coextension
of scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit' : coextend_scalars f ⋙ restrict_scalars f ⟶ 𝟭 (Module R) | { app := λ X,
{ to_fun := λ g, g.to_fun (1 : S),
map_add' := λ x1 x2, by simp [linear_map.to_fun_eq_coe],
map_smul' := λ r (g : (restrict_scalars f).obj ((coextend_scalars f).obj X)),
begin
simp only [linear_map.to_fun_eq_coe, ring_hom.id_apply],
rw [restrict_scalars.smul_def f, coextend_scala... | def | category_theory.Module.restriction_coextension_adj.counit' | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"linear_map.ext",
"linear_map.to_fun_eq_coe",
"mul_one",
"one_mul",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_eq_mul"
] | The natural transformation from the composition of coextension and restriction of scalars to
identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_coextend_scalars_adj {R : Type u₁} {S : Type u₂} [ring R] [ring S]
(f : R →+* S) : restrict_scalars f ⊣ coextend_scalars f | { hom_equiv := λ X Y,
{ to_fun := restriction_coextension_adj.hom_equiv.from_restriction f,
inv_fun := restriction_coextension_adj.hom_equiv.to_restriction f,
left_inv := λ g, linear_map.ext $ λ (x : X), by simp,
right_inv := λ g, linear_map.ext $ λ x, linear_map.ext $ λ (s : S), by simp },
unit := rest... | def | category_theory.Module.restrict_coextend_scalars_adj | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"inv_fun",
"linear_map.ext",
"restrict_scalars",
"ring"
] | Restriction of scalars is left adjoint to coextension of scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv.to_restrict_scalars {X Y} (g : (extend_scalars f).obj X ⟶ Y) :
X ⟶ (restrict_scalars f).obj Y | { to_fun := λ x, g $ (1 : S) ⊗ₜ[R, f] x,
map_add' := λ _ _, by rw [tmul_add, map_add],
map_smul' := λ r x,
begin
letI : module R S := module.comp_hom S f,
letI : module R Y := module.comp_hom Y f,
rw [ring_hom.id_apply, restrict_scalars.smul_def, ←linear_map.map_smul, tmul_smul],
congr,
end } | def | category_theory.Module.extend_restrict_scalars_adj.hom_equiv.to_restrict_scalars | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"module",
"module.comp_hom",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply"
] | Given `R`-module X and `S`-module Y and a map `g : (extend_scalars f).obj X ⟶ Y`, i.e. `S`-linear
map `S ⨂ X → Y`, there is a `X ⟶ (restrict_scalars f).obj Y`, i.e. `R`-linear map `X ⟶ Y` by
`x ↦ g (1 ⊗ x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv.from_extend_scalars {X Y} (g : X ⟶ (restrict_scalars f).obj Y) :
(extend_scalars f).obj X ⟶ Y | begin
letI m1 : module R S := module.comp_hom S f, letI m2 : module R Y := module.comp_hom Y f,
refine ⟨λ z, tensor_product.lift ⟨λ s, ⟨_, _, _⟩, _, _⟩ z, _, _⟩,
{ exact λ x, s • g x },
{ intros, rw [map_add, smul_add], },
{ intros, rw [ring_hom.id_apply, smul_comm, ←linear_map.map_smul], },
{ intros, ext, ... | def | category_theory.Module.extend_restrict_scalars_adj.hom_equiv.from_extend_scalars | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"linear_map.add_apply",
"linear_map.coe_mk",
"linear_map.smul_apply",
"module",
"module.comp_hom",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_add",
"smul_eq_mul",
"smul_zero",
"tensor_product.induction_on",
"tensor_product.lift"
] | Given `R`-module X and `S`-module Y and a map `X ⟶ (restrict_scalars f).obj Y`, i.e `R`-linear map
`X ⟶ Y`, there is a map `(extend_scalars f).obj X ⟶ Y`, i.e `S`-linear map `S ⨂ X → Y` by
`s ⊗ x ↦ s • g x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv {X Y} : ((extend_scalars f).obj X ⟶ Y) ≃ (X ⟶ (restrict_scalars f).obj Y) | { to_fun := hom_equiv.to_restrict_scalars f,
inv_fun := hom_equiv.from_extend_scalars f,
left_inv := λ g, begin
ext z,
induction z using tensor_product.induction_on with x s z1 z2 ih1 ih2,
{ simp only [map_zero], },
{ erw tensor_product.lift.tmul,
simp only [linear_map.coe_mk],
change S ... | def | category_theory.Module.extend_restrict_scalars_adj.hom_equiv | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"inv_fun",
"linear_map.coe_mk",
"mul_one",
"one_smul",
"restrict_scalars",
"tensor_product.induction_on",
"tensor_product.lift.tmul"
] | Given `R`-module X and `S`-module Y, `S`-linear linear maps `(extend_scalars f).obj X ⟶ Y`
bijectively correspond to `R`-linear maps `X ⟶ (restrict_scalars f).obj Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit.map {X} : X ⟶ (extend_scalars f ⋙ restrict_scalars f).obj X | { to_fun := λ x, (1 : S) ⊗ₜ[R, f] x,
map_add' := λ x x', by { rw tensor_product.tmul_add, },
map_smul' := λ r x, by { letI m1 : module R S := module.comp_hom S f, tidy } } | def | category_theory.Module.extend_restrict_scalars_adj.unit.map | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"module",
"module.comp_hom",
"restrict_scalars",
"tensor_product.tmul_add"
] | For any `R`-module X, there is a natural `R`-linear map from `X` to `X ⨂ S` by sending `x ↦ x ⊗ 1` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit : 𝟭 (Module R) ⟶ extend_scalars f ⋙ restrict_scalars f | { app := λ _, unit.map f, naturality' := λ X X' g, by tidy } | def | category_theory.Module.extend_restrict_scalars_adj.unit | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"restrict_scalars"
] | The natural transformation from identity functor on `R`-module to the composition of extension and
restriction of scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit.map {Y} : (restrict_scalars f ⋙ extend_scalars f).obj Y ⟶ Y | begin
letI m1 : module R S := module.comp_hom S f,
letI m2 : module R Y := module.comp_hom Y f,
refine ⟨tensor_product.lift ⟨λ (s : S), ⟨λ (y : Y), s • y, smul_add _, _⟩, _, _⟩, _, _⟩,
{ intros, rw [ring_hom.id_apply, restrict_scalars.smul_def, ←mul_smul, mul_comm, mul_smul,
restrict_scalars.smul_def], },... | def | category_theory.Module.extend_restrict_scalars_adj.counit.map | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"add_smul",
"linear_map.add_apply",
"linear_map.coe_mk",
"linear_map.smul_apply",
"module",
"module.comp_hom",
"mul_comm",
"restrict_scalars",
"restrict_scalars.smul_def",
"ring_hom.id_apply",
"smul_add",
"smul_eq_mul",
"smul_zero",
"tensor_product.induction_on",
"tensor_product.lift.tmu... | For any `S`-module Y, there is a natural `R`-linear map from `S ⨂ Y` to `Y` by
`s ⊗ y ↦ s • y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit : (restrict_scalars f ⋙ extend_scalars f) ⟶ (𝟭 (Module S)) | { app := λ _, counit.map f,
naturality' := λ Y Y' g,
begin
ext z, induction z using tensor_product.induction_on,
{ simp only [map_zero] },
{ simp only [category_theory.functor.comp_map, Module.coe_comp, function.comp_app,
extend_scalars.map_tmul, restrict_scalars.map_apply, counit.map_... | def | category_theory.Module.extend_restrict_scalars_adj.counit | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module",
"Module.coe_comp",
"category_theory.functor.comp_map",
"category_theory.functor.id_map",
"linear_map.coe_mk",
"linear_map.map_smulₛₗ",
"restrict_scalars",
"ring_hom.id_apply",
"tensor_product.induction_on"
] | The natural transformation from the composition of restriction and extension of scalars to the
identity functor on `S`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_restrict_scalars_adj {R : Type u₁} {S : Type u₂} [comm_ring R] [comm_ring S]
(f : R →+* S) : extend_scalars f ⊣ restrict_scalars f | { hom_equiv := λ _ _, extend_restrict_scalars_adj.hom_equiv f,
unit := extend_restrict_scalars_adj.unit f,
counit := extend_restrict_scalars_adj.counit f,
hom_equiv_unit' := λ X Y g, linear_map.ext $ λ x, by simp,
hom_equiv_counit' := λ X Y g, linear_map.ext $ λ x,
begin
induction x using tensor_produ... | def | category_theory.Module.extend_restrict_scalars_adj | algebra.category.Module | src/algebra/category/Module/change_of_rings.lean | [
"algebra.category.Module.basic",
"ring_theory.tensor_product"
] | [
"Module.coe_comp",
"comm_ring",
"linear_map.coe_mk",
"linear_map.ext",
"restrict_scalars",
"tensor_product.induction_on",
"tensor_product.lift.tmul"
] | Given commutative rings `R, S` and a ring hom `f : R →+* S`, the extension and restriction of
scalars by `f` are adjoint to each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prequotient
-- There's always `of`
| of : Π (j : J) (x : F.obj j), prequotient
-- Then one generator for each operation
| zero : prequotient
| neg : prequotient → prequotient
| add : prequotient → prequotient → prequotient
| smul : R → prequotient → prequotient | inductive | Module.colimits.prequotient | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | An inductive type representing all module expressions (without relations)
on a collection of types indexed by the objects of `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relation : prequotient F → prequotient F → Prop
-- Make it an equivalence relation:
| refl : Π (x), relation x x
| symm : Π (x y) (h : relation x y), relation y x
| trans : Π (x y z) (h : relation x y) (k : relation y z), relation x z
-- There's always a `map` relation
| map : Π (j j' : J) (f : j ⟶ j') (x : F.obj j), r... | inductive | Module.colimits.relation | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [
"add_smul",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | The relation on `prequotient` saying when two expressions are equal
because of the module laws, or
because one element is mapped to another by a morphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_type : Type (max u v w) | quotient (colimit_setoid F) | def | Module.colimits.colimit_type | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | The underlying type of the colimit of a diagram in `Module R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_smul (s x) :
quot.mk setoid.r (smul s x) = (s • (quot.mk setoid.r x) : colimit_type F) | rfl | lemma | Module.colimits.quot_smul | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : Module R | Module.of R (colimit_type F) | def | Module.colimits.colimit | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [
"Module",
"Module.of"
] | The bundled module giving the colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (j : J) : F.obj j ⟶ colimit F | { to_fun := cocone_fun F j,
map_smul' := by { intros, apply quot.sound, apply relation.smul, },
map_add' := by intros; apply quot.sound; apply relation.add } | def | Module.colimits.cocone_morphism | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | The group homomorphism from a given module in the diagram to the colimit module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(cocone_morphism F j') (F.map f x) = (cocone_morphism F j) x | by { rw ←cocone_naturality F f, refl } | lemma | Module.colimits.cocone_naturality_components | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc_fun_lift (s : cocone F) : prequotient F → s.X | | (of j x) := (s.ι.app j) x
| zero := 0
| (neg x) := -(desc_fun_lift x)
| (add x y) := desc_fun_lift x + desc_fun_lift y
| (smul s x) := s • (desc_fun_lift x) | def | Module.colimits.desc_fun_lift | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | The function from the free module on the diagram to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_fun (s : cocone F) : colimit_type F → s.X | begin
fapply quot.lift,
{ exact desc_fun_lift F s },
{ intros x y r,
induction r; try { dsimp },
-- refl
{ refl },
-- symm
{ exact r_ih.symm },
-- trans
{ exact eq.trans r_ih_h r_ih_k },
-- map
{ simp, },
-- zero
{ simp, },
-- neg
{ simp, },
-- add
{ sim... | def | Module.colimits.desc_fun | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [
"add_smul",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | The function from the colimit module to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_morphism (s : cocone F) : colimit F ⟶ s.X | { to_fun := desc_fun F s,
map_smul' := λ s x, by { induction x; refl, },
map_add' := λ x y, by { induction x; induction y; refl }, } | def | Module.colimits.desc_morphism | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | The group homomorphism from the colimit module to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit (colimit_cocone F) | { desc := λ s, desc_morphism F s,
uniq' := λ s m w,
begin
ext,
induction x,
induction x,
{ have w' := congr_fun (congr_arg (λ f : F.obj x_j ⟶ s.X, (f : F.obj x_j → s.X)) (w x_j)) x_x,
erw w',
refl, },
{ simp *, },
{ simp *, },
{ simp *, },
{ simp *, },
refl
end }. | def | Module.colimits.colimit_cocone_is_colimit | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | Evidence that the proposed colimit is the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_Module : has_colimits (Module.{max v u} R) | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk
{ cocone := colimit_cocone F,
is_colimit := colimit_cocone_is_colimit F } } } | instance | Module.colimits.has_colimits_Module | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_size_Module : has_colimits_of_size.{v} (Module.{max v u} R) | has_colimits_of_size_shrink _ | instance | Module.colimits.has_colimits_of_size_Module | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_size_zero_Module : has_colimits_of_size.{0} (Module.{max v u} R) | @has_colimits_of_size_shrink.{0} (Module.{max v u} R) _ Module.colimits.has_colimits_Module | instance | Module.colimits.has_colimits_of_size_zero_Module | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [
"Module.colimits.has_colimits_Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_Module' (R : Type u) [ring R] :
has_colimits (Module.{max u v} R) | Module.colimits.has_colimits_Module.{u v} | instance | Module.colimits.has_colimits_Module' | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_Module'' (R : Type u) [ring R] :
has_colimits (Module.{u} R) | Module.colimits.has_colimits_Module.{u u} | instance | Module.colimits.has_colimits_Module'' | algebra.category.Module | src/algebra/category/Module/colimits.lean | [
"algebra.category.Module.basic",
"category_theory.concrete_category.elementwise"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_bot_of_mono [mono f] : f.ker = ⊥ | linear_map.ker_eq_bot_of_cancel $ λ u v, (@cancel_mono _ _ _ _ _ f _ ↟u ↟v).1 | lemma | Module.ker_eq_bot_of_mono | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"linear_map.ker_eq_bot_of_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_top_of_epi [epi f] : f.range = ⊤ | linear_map.range_eq_top_of_cancel $ λ u v, (@cancel_epi _ _ _ _ _ f _ ↟u ↟v).1 | lemma | Module.range_eq_top_of_epi | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"linear_map.range_eq_top_of_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_ker_eq_bot : mono f ↔ f.ker = ⊥ | ⟨λ hf, by exactI ker_eq_bot_of_mono _,
λ hf, concrete_category.mono_of_injective _ $ linear_map.ker_eq_bot.1 hf⟩ | lemma | Module.mono_iff_ker_eq_bot | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_injective : mono f ↔ function.injective f | by rw [mono_iff_ker_eq_bot, linear_map.ker_eq_bot] | lemma | Module.mono_iff_injective | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"linear_map.ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_range_eq_top : epi f ↔ f.range = ⊤ | ⟨λ hf, by exactI range_eq_top_of_epi _,
λ hf, concrete_category.epi_of_surjective _ $ linear_map.range_eq_top.1 hf⟩ | lemma | Module.epi_iff_range_eq_top | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_surjective : epi f ↔ function.surjective f | by rw [epi_iff_range_eq_top, linear_map.range_eq_top] | lemma | Module.epi_iff_surjective | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"linear_map.range_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_of_epi_zero (X) [h : epi (0 : X ⟶ of R M)] : unique M | unique_of_surjective_zero X ((Module.epi_iff_surjective _).mp h) | def | Module.unique_of_epi_zero | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"Module.epi_iff_surjective",
"unique"
] | If the zero morphism is an epi then the codomain is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_as_hom'_subtype (U : submodule R X) : mono ↾U.subtype | (mono_iff_ker_eq_bot _).mpr (submodule.ker_subtype U) | instance | Module.mono_as_hom'_subtype | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"submodule",
"submodule.ker_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_as_hom''_mkq (U : submodule R X) : epi ↿U.mkq | (epi_iff_range_eq_top _).mpr $ submodule.range_mkq _ | instance | Module.epi_as_hom''_mkq | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"submodule",
"submodule.range_mkq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_epimorphisms : (forget (Module.{v} R)).preserves_epimorphisms | { preserves := λ X Y f hf, by rwa [forget_map_eq_coe, category_theory.epi_iff_surjective,
← epi_iff_surjective] } | instance | Module.forget_preserves_epimorphisms | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"category_theory.epi_iff_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_monomorphisms : (forget (Module.{v} R)).preserves_monomorphisms | { preserves := λ X Y f hf, by rwa [forget_map_eq_coe, category_theory.mono_iff_injective,
← mono_iff_injective] } | instance | Module.forget_preserves_monomorphisms | algebra.category.Module | src/algebra/category/Module/epi_mono.lean | [
"linear_algebra.quotient",
"algebra.category.Module.basic"
] | [
"category_theory.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
M : AddCommGroup | AddCommGroup.filtered_colimits.colimit (F ⋙ forget₂ (Module R) AddCommGroup.{max v u}) | abbreviation | Module.filtered_colimits.M | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module"
] | The colimit of `F ⋙ forget₂ (Module R) AddCommGroup` in the category `AddCommGroup`.
In the following, we will show that this has the structure of an `R`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
M.mk : (Σ j, F.obj j) → M | quot.mk (types.quot.rel (F ⋙ forget (Module R))) | abbreviation | Module.filtered_colimits.M.mk | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module"
] | The canonical projection into the colimit, as a quotient type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
M.mk_eq (x y : Σ j, F.obj j)
(h : ∃ (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) :
M.mk x = M.mk y | quot.eqv_gen_sound (types.filtered_colimit.eqv_gen_quot_rel_of_rel (F ⋙ forget (Module R)) x y h) | lemma | Module.filtered_colimits.M.mk_eq | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_smul_aux (r : R) (x : Σ j, F.obj j) : M | M.mk ⟨x.1, r • x.2⟩ | def | Module.filtered_colimits.colimit_smul_aux | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [] | The "unlifted" version of scalar multiplication in the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_smul_aux_eq_of_rel (r : R) (x y : Σ j, F.obj j)
(h : types.filtered_colimit.rel (F ⋙ forget (Module R)) x y) :
colimit_smul_aux r x = colimit_smul_aux r y | begin
apply M.mk_eq,
obtain ⟨k, f, g, hfg⟩ := h,
use [k, f, g],
simp only [category_theory.functor.comp_map, forget_map_eq_coe] at hfg,
rw [linear_map.map_smul, linear_map.map_smul, hfg],
end | lemma | Module.filtered_colimits.colimit_smul_aux_eq_of_rel | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module",
"category_theory.functor.comp_map",
"linear_map.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_has_smul : has_smul R M | { smul := λ r x, begin
refine quot.lift (colimit_smul_aux F r) _ x,
intros x y h,
apply colimit_smul_aux_eq_of_rel,
apply types.filtered_colimit.rel_of_quot_rel,
exact h,
end } | instance | Module.filtered_colimits.colimit_has_smul | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"has_smul"
] | Scalar multiplication in the colimit. See also `colimit_smul_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_smul_mk_eq (r : R) (x : Σ j, F.obj j) : r • M.mk x = M.mk ⟨x.1, r • x.2⟩ | rfl | lemma | Module.filtered_colimits.colimit_smul_mk_eq | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_module : module R M | { one_smul := λ x, begin
apply quot.induction_on x, clear x, intro x, cases x with j x,
erw [colimit_smul_mk_eq F 1 ⟨j, x⟩, one_smul],
refl,
end,
mul_smul := λ r s x, begin
apply quot.induction_on x, clear x, intro x, cases x with j x,
erw [colimit_smul_mk_eq F (r * s) ⟨j, x⟩, colimit_smul_mk_eq... | instance | Module.filtered_colimits.colimit_module | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"add_smul",
"linear_map.map_smul",
"module",
"one_smul",
"quot.induction_on₂",
"smul_add",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : Module R | Module.of R M | def | Module.filtered_colimits.colimit | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module",
"Module.of"
] | The bundled `R`-module giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (j : J) : F.obj j ⟶ colimit | { map_smul' := λ r x, begin erw colimit_smul_mk_eq F r ⟨j, x⟩, refl, end,
.. (AddCommGroup.filtered_colimits.colimit_cocone
(F ⋙ forget₂ (Module R) AddCommGroup.{max v u})).ι.app j } | def | Module.filtered_colimits.cocone_morphism | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module"
] | The linear map from a given `R`-module in the diagram to the colimit module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι :=
{ app := cocone_morphism,
naturality' := λ j j' f,
linear_map.coe_injective ((types.colimit_cocone (F ⋙ forget (Module R))).ι.naturality f) } } | def | Module.filtered_colimits.colimit_cocone | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module",
"linear_map.coe_injective"
] | The cocone over the proposed colimit module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_desc (t : cocone F) : colimit ⟶ t.X | { map_smul' := λ r x, begin
apply quot.induction_on x, clear x, intro x, cases x with j x,
erw colimit_smul_mk_eq,
exact linear_map.map_smul (t.ι.app j) r x,
end,
.. (AddCommGroup.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ (Module R) AddCommGroup.{max v u})).desc
((forget₂ (Mod... | def | Module.filtered_colimits.colimit_desc | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module",
"linear_map.map_smul"
] | Given a cocone `t` of `F`, the induced monoid linear map from the colimit to the cocone point.
We already know that this is a morphism between additive groups. The only thing left to see is that
it is a linear map, i.e. preserves scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := colimit_desc,
fac' := λ t j, linear_map.coe_injective $
(types.colimit_cocone_is_colimit (F ⋙ forget (Module R))).fac
((forget (Module R)).map_cocone t) j,
uniq' := λ t m h, linear_map.coe_injective $
(types.colimit_cocone_is_colimit (F ⋙ forget (Module R))).uniq
((forget (Module R)).map_c... | def | Module.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module",
"linear_map.coe_injective",
"linear_map.congr_fun"
] | The proposed colimit cocone is a colimit in `Module R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommGroup_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ (Module R) AddCommGroup.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit F)
(AddCommGroup.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ (Module.{u} R) AddCommGroup.{u})) } } | instance | Module.filtered_colimits.forget₂_AddCommGroup_preserves_filtered_colimits | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget (Module.{u} R)) | limits.comp_preserves_filtered_colimits (forget₂ (Module R) AddCommGroup) (forget AddCommGroup) | instance | Module.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Module | src/algebra/category/Module/filtered_colimits.lean | [
"algebra.category.Group.filtered_colimits",
"algebra.category.Module.basic"
] | [
"Module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image : Module R | Module.of R (linear_map.range f) | def | Module.image | algebra.category.Module | src/algebra/category/Module/images.lean | [
"algebra.category.Module.abelian",
"category_theory.limits.shapes.images"
] | [
"Module",
"Module.of",
"linear_map.range"
] | The image of a morphism in `Module R` is just the bundling of `linear_map.range f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.lift (F' : mono_factorisation f) : image f ⟶ F'.I | { to_fun :=
(λ x, F'.e (classical.indefinite_description _ x.2).1 : image f → F'.I),
map_add' :=
begin
intros x y,
haveI := F'.m_mono,
apply (mono_iff_injective F'.m).1, apply_instance,
rw [linear_map.map_add],
change (F'.e ≫ F'.m) _ = (F'.e ≫ F'.m) _ + (F'.e ≫ F'.m) _,
rw [F'.fac],
rw... | def | Module.image.lift | algebra.category.Module | src/algebra/category/Module/images.lean | [
"algebra.category.Module.abelian",
"category_theory.limits.shapes.images"
] | [
"linear_map.map_add",
"linear_map.map_smul"
] | The universal property for the image factorisation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_iso_range {G H : Module.{v} R} (f : G ⟶ H) :
limits.image f ≅ Module.of R f.range | is_image.iso_ext (image.is_image f) (is_image f) | def | Module.image_iso_range | algebra.category.Module | src/algebra/category/Module/images.lean | [
"algebra.category.Module.abelian",
"category_theory.limits.shapes.images"
] | [
"Module.of"
] | The categorical image of a morphism in `Module R`
agrees with the linear algebraic range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_iso_range_inv_image_ι {G H : Module.{v} R} (f : G ⟶ H) :
(image_iso_range f).inv ≫ limits.image.ι f = Module.of_hom f.range.subtype | is_image.iso_ext_inv_m _ _ | lemma | Module.image_iso_range_inv_image_ι | algebra.category.Module | src/algebra/category/Module/images.lean | [
"algebra.category.Module.abelian",
"category_theory.limits.shapes.images"
] | [
"Module.of_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_iso_range_hom_subtype {G H : Module.{v} R} (f : G ⟶ H) :
(image_iso_range f).hom ≫ Module.of_hom f.range.subtype = limits.image.ι f | by erw [←image_iso_range_inv_image_ι f, iso.hom_inv_id_assoc] | lemma | Module.image_iso_range_hom_subtype | algebra.category.Module | src/algebra/category/Module/images.lean | [
"algebra.category.Module.abelian",
"category_theory.limits.shapes.images"
] | [
"Module.of_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_cone : kernel_fork f | kernel_fork.of_ι (as_hom f.ker.subtype) $ by tidy | def | Module.kernel_cone | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [] | The kernel cone induced by the concrete kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_is_limit : is_limit (kernel_cone f) | fork.is_limit.mk _
(λ s, linear_map.cod_restrict f.ker (fork.ι s) (λ c, linear_map.mem_ker.2 $
by { rw [←@function.comp_apply _ _ _ f (fork.ι s) c, ←coe_comp, fork.condition,
has_zero_morphisms.comp_zero (fork.ι s) N], refl }))
(λ s, linear_map.subtype_comp_cod_restrict _ _ _)
(λ s m h, linear_map.ext $... | def | Module.kernel_is_limit | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"function.comp_apply",
"linear_map.cod_restrict",
"linear_map.ext",
"linear_map.subtype_comp_cod_restrict"
] | The kernel of a linear map is a kernel in the categorical sense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cocone : cokernel_cofork f | cokernel_cofork.of_π (as_hom f.range.mkq) $ linear_map.range_mkq_comp _ | def | Module.cokernel_cocone | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"linear_map.range_mkq_comp"
] | The cokernel cocone induced by the projection onto the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_is_colimit : is_colimit (cokernel_cocone f) | cofork.is_colimit.mk _
(λ s, f.range.liftq (cofork.π s) $ linear_map.range_le_ker_iff.2 $ cokernel_cofork.condition s)
(λ s, f.range.liftq_mkq (cofork.π s) _)
(λ s m h,
begin
haveI : epi (as_hom f.range.mkq) := (epi_iff_range_eq_top _).mpr (submodule.range_mkq _),
apply (cancel_epi (as_hom f.range.mkq))... | def | Module.cokernel_is_colimit | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"submodule.liftq_mkq",
"submodule.range_mkq"
] | The projection onto the quotient is a cokernel in the categorical sense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_kernels_Module : has_kernels (Module R) | ⟨λ X Y f, has_limit.mk ⟨_, kernel_is_limit f⟩⟩ | lemma | Module.has_kernels_Module | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"Module"
] | The category of R-modules has kernels, given by the inclusion of the kernel submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cokernels_Module : has_cokernels (Module R) | ⟨λ X Y f, has_colimit.mk ⟨_, cokernel_is_colimit f⟩⟩ | lemma | Module.has_cokernels_Module | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"Module"
] | The category or R-modules has cokernels, given by the projection onto the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_iso_ker {G H : Module.{v} R} (f : G ⟶ H) :
kernel f ≅ Module.of R (f.ker) | limit.iso_limit_cone ⟨_, kernel_is_limit f⟩ | def | Module.kernel_iso_ker | algebra.category.Module | src/algebra/category/Module/kernels.lean | [
"algebra.category.Module.epi_mono",
"category_theory.concrete_category.elementwise"
] | [
"Module.of"
] | The categorical kernel of a morphism in `Module`
agrees with the usual module-theoretical kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_iso_ker_inv_kernel_ι :
(kernel_iso_ker f).inv ≫ kernel.ι f = f.ker.subtype | limit.iso_limit_cone_inv_π _ _ | lemma | Module.kernel_iso_ker_inv_kernel_ι | 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 | |
kernel_iso_ker_hom_ker_subtype :
(kernel_iso_ker f).hom ≫ f.ker.subtype = kernel.ι f | is_limit.cone_point_unique_up_to_iso_inv_comp _ (limit.is_limit _) walking_parallel_pair.zero | lemma | Module.kernel_iso_ker_hom_ker_subtype | 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 |
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