The category of R-modules has finite biproducts #
Construct limit data for a binary product in ModuleCat R, using ModuleCat.of R (M × N).
Equations
- One or more equations did not get rendered due to their size.
@[simp]
theorem
ModuleCat.binaryProductLimitCone_isLimit_lift
{R : Type u}
[Ring R]
(M N : ModuleCat R)
(s : CategoryTheory.Limits.Cone (CategoryTheory.Limits.pair M N))
:
(M.binaryProductLimitCone N).isLimit.lift s = ofHom
((Hom.hom (s.π.app { as := CategoryTheory.Limits.WalkingPair.left })).prod
(Hom.hom (s.π.app { as := CategoryTheory.Limits.WalkingPair.right })))
@[simp]
theorem
ModuleCat.binaryProductLimitCone_cone_π_app_left
{R : Type u}
[Ring R]
(M N : ModuleCat R)
:
(M.binaryProductLimitCone N).cone.π.app { as := CategoryTheory.Limits.WalkingPair.left } = ofHom (LinearMap.fst R ↑M ↑N)
@[simp]
theorem
ModuleCat.binaryProductLimitCone_cone_π_app_right
{R : Type u}
[Ring R]
(M N : ModuleCat R)
:
(M.binaryProductLimitCone N).cone.π.app { as := CategoryTheory.Limits.WalkingPair.right } = ofHom (LinearMap.snd R ↑M ↑N)
We verify that the biproduct in ModuleCat R is isomorphic to
the cartesian product of the underlying types:
Equations
@[simp]
theorem
ModuleCat.biprodIsoProd_inv_comp_fst_apply
{R : Type u}
[Ring R]
(M N : ModuleCat R)
(x : CategoryTheory.ToType (of R (↑M × ↑N)))
:
@[simp]
theorem
ModuleCat.biprodIsoProd_inv_comp_snd_apply
{R : Type u}
[Ring R]
(M N : ModuleCat R)
(x : CategoryTheory.ToType (of R (↑M × ↑N)))
:
def
ModuleCat.HasLimit.lift
{R : Type u}
[Ring R]
{J : Type w}
(f : J → ModuleCat R)
(s : CategoryTheory.Limits.Fan f)
:
The map from an arbitrary cone over an indexed family of abelian groups to the cartesian product of those groups.
Equations
- ModuleCat.HasLimit.lift f s = ModuleCat.ofHom { toFun := fun (x : ↑s.1) (j : J) => (CategoryTheory.ConcreteCategory.hom (s.π.app { as := j })) x, map_add' := ⋯, map_smul' := ⋯ }
@[simp]
theorem
ModuleCat.HasLimit.lift_hom_apply
{R : Type u}
[Ring R]
{J : Type w}
(f : J → ModuleCat R)
(s : CategoryTheory.Limits.Fan f)
(x : ↑s.1)
(j : J)
:
Construct limit data for a product in ModuleCat R, using ModuleCat.of R (∀ j, F.obj j).
Equations
- One or more equations did not get rendered due to their size.
@[simp]
theorem
ModuleCat.HasLimit.productLimitCone_isLimit_lift
{R : Type u}
[Ring R]
{J : Type w}
(f : J → ModuleCat R)
(s : CategoryTheory.Limits.Fan f)
:
@[simp]
theorem
ModuleCat.HasLimit.productLimitCone_cone_pt_isAddCommGroup
{R : Type u}
[Ring R]
{J : Type w}
(f : J → ModuleCat R)
:
@[simp]
theorem
ModuleCat.HasLimit.productLimitCone_cone_π
{R : Type u}
[Ring R]
{J : Type w}
(f : J → ModuleCat R)
:
(productLimitCone f).cone.π = CategoryTheory.Discrete.natTrans fun (j : CategoryTheory.Discrete J) => ofHom (LinearMap.proj j.as)
@[simp]
theorem
ModuleCat.HasLimit.productLimitCone_cone_pt_carrier
{R : Type u}
[Ring R]
{J : Type w}
(f : J → ModuleCat R)
:
theorem
ModuleCat.biproductIsoPi_inv_comp_π_apply
{R : Type u}
[Ring R]
{J : Type}
[Finite J]
(f : J → ModuleCat R)
(j : J)
(x : CategoryTheory.ToType (of R ((j : J) → ↑(f j))))
:
noncomputable def
lequivProdOfRightSplitExact
{R : Type u}
{A M B : Type v}
[Ring R]
[AddCommGroup A]
[Module R A]
[AddCommGroup B]
[Module R B]
[AddCommGroup M]
[Module R M]
{j : A →ₗ[R] M}
{g : M →ₗ[R] B}
{f : B →ₗ[R] M}
(hj : Function.Injective ⇑j)
(exac : LinearMap.range j = LinearMap.ker g)
(h : g ∘ₗ f = LinearMap.id)
:
The isomorphism A × B ≃ₗ[R] M coming from a right split exact sequence 0 ⟶ A ⟶ M ⟶ B ⟶ 0
of modules.
Equations
- One or more equations did not get rendered due to their size.
noncomputable def
lequivProdOfLeftSplitExact
{R : Type u}
{A M B : Type v}
[Ring R]
[AddCommGroup A]
[Module R A]
[AddCommGroup B]
[Module R B]
[AddCommGroup M]
[Module R M]
{j : A →ₗ[R] M}
{g : M →ₗ[R] B}
{f : M →ₗ[R] A}
(hg : Function.Surjective ⇑g)
(exac : LinearMap.range j = LinearMap.ker g)
(h : f ∘ₗ j = LinearMap.id)
:
The isomorphism A × B ≃ₗ[R] M coming from a left split exact sequence 0 ⟶ A ⟶ M ⟶ B ⟶ 0
of modules.
Equations
- One or more equations did not get rendered due to their size.