File: Mathlib/Algebra/Category/Grp/Adjunctions.lean

full_name: AddCommGrp.free
code: /-- The free functor `Type u ⥤ AddCommGroup` sending a type `X` to the
free abelian group with generators `x : X`.
-/
def free : Type u ⥤ AddCommGrp 
goal: Type u ⥤ AddCommGrp

full_name: AddCommGrp.free_obj_coe
code: @[simp]
theorem free_obj_coe {α : Type u} : (free.obj α : Type u) = FreeAbelianGroup α 
goal: ↑(AddCommGrp.free.obj α) = FreeAbelianGroup α

full_name: AddCommGrp.free_map_coe
code: theorem free_map_coe {α β : Type u} {f : α → β} (x : FreeAbelianGroup α) :
    (free.map f) x = f <$> x 
goal: (AddCommGrp.free.map f) x = f <$> x

full_name: AddCommGrp.adj
code: /-- The free-forgetful adjunction for abelian groups.
-/
def adj : free ⊣ forget AddCommGrp.{u} 
goal: AddCommGrp.free ⊣ CategoryTheory.forget AddCommGrp

full_name: AddCommGrp.instIsLeftAdjointFree
code: instance : free.{u}.IsLeftAdjoint 
goal: AddCommGrp.free.IsLeftAdjoint

full_name: AddCommGrp.instIsRightAdjointForget
code: instance : (forget AddCommGrp.{u}).IsRightAdjoint 
goal: (CategoryTheory.forget AddCommGrp).IsRightAdjoint

full_name: AddCommGrp.instPreservesMonomorphismsFree
code: instance : (free.{u}).PreservesMonomorphisms 
goal: AddCommGrp.free.PreservesMonomorphisms

full_name: Grp.free
code: /-- The free functor `Type u ⥤ Group` sending a type `X` to the free group with generators `x : X`.
-/
def free : Type u ⥤ Grp 
goal: Type u ⥤ Grp

full_name: Grp.adj
code: /-- The free-forgetful adjunction for groups.
-/
def adj : free ⊣ forget Grp.{u} 
goal: Grp.free ⊣ CategoryTheory.forget Grp

full_name: Grp.instIsRightAdjointForget
code: instance : (forget Grp.{u}).IsRightAdjoint  
goal: (CategoryTheory.forget Grp).IsRightAdjoint

full_name: Grp.abelianize
code: /-- The abelianization functor `Group ⥤ CommGroup` sending a group `G` to its abelianization `Gᵃᵇ`.
 -/
def abelianize : Grp.{u} ⥤ CommGrp.{u} 
goal: Grp ⥤ CommGrp

full_name: Grp.abelianizeAdj
code: /-- The abelianization-forgetful adjuction from `Group` to `CommGroup`. -/
def abelianizeAdj : abelianize ⊣ forget₂ CommGrp.{u} Grp.{u} 
goal: Grp.abelianize ⊣ CategoryTheory.forget₂ CommGrp Grp

full_name: MonCat.units
code: /-- The functor taking a monoid to its subgroup of units. -/
@[simps]
def MonCat.units : MonCat.{u} ⥤ Grp.{u} 
goal: MonCat ⥤ Grp

full_name: MonCat.units_map
code: /-- The functor taking a monoid to its subgroup of units. -/
@[simps]
def MonCat.units : MonCat.{u} ⥤ Grp.{u} 
goal: ∀ {X Y : MonCat} (f : X ⟶ Y), MonCat.units.map f = Grp.ofHom (Units.map f)

full_name: MonCat.units_obj
code: /-- The functor taking a monoid to its subgroup of units. -/
@[simps]
def MonCat.units : MonCat.{u} ⥤ Grp.{u} 
goal: MonCat.units.obj R = Grp.of (↑R)ˣ

full_name: Grp.forget₂MonAdj
code: /-- The forgetful-units adjunction between `Grp` and `MonCat`. -/
def Grp.forget₂MonAdj : forget₂ Grp MonCat ⊣ MonCat.units.{u} 
goal: CategoryTheory.forget₂ Grp MonCat ⊣ MonCat.units

full_name: instIsRightAdjointGrpMonCatUnits
code: instance : MonCat.units.{u}.IsRightAdjoint 
goal: MonCat.units.IsRightAdjoint

full_name: CommMonCat.units_map
code: /-- The functor taking a monoid to its subgroup of units. -/
@[simps]
def CommMonCat.units : CommMonCat.{u} ⥤ CommGrp.{u} 
goal: ∀ {X Y : CommMonCat} (f : X ⟶ Y), CommMonCat.units.map f = CommGrp.ofHom (Units.map f)

full_name: CommMonCat.units
code: /-- The functor taking a monoid to its subgroup of units. -/
@[simps]
def CommMonCat.units : CommMonCat.{u} ⥤ CommGrp.{u} 
goal: CommMonCat ⥤ CommGrp

full_name: CommMonCat.units_obj
code: /-- The functor taking a monoid to its subgroup of units. -/
@[simps]
def CommMonCat.units : CommMonCat.{u} ⥤ CommGrp.{u} 
goal: CommMonCat.units.obj R = CommGrp.of (↑R)ˣ

full_name: CommGrp.forget₂CommMonAdj
code: /-- The forgetful-units adjunction between `CommGrp` and `CommMonCat`. -/
def CommGrp.forget₂CommMonAdj : forget₂ CommGrp CommMonCat ⊣ CommMonCat.units.{u} 
goal: CategoryTheory.forget₂ CommGrp CommMonCat ⊣ CommMonCat.units

full_name: instIsRightAdjointCommGrpCommMonCatUnits
code: instance : CommMonCat.units.{u}.IsRightAdjoint 
goal: CommMonCat.units.IsRightAdjoint