File: Mathlib/Algebra/Category/Ring/FilteredColimits.lean

full_name: SemiRingCat.FilteredColimits.semiringObj
code: instance semiringObj (j : J) :
    Semiring (((F ⋙ forget₂ SemiRingCatMax.{v, u} MonCat) ⋙ forget MonCat).obj j) 
goal: Semiring (((F ⋙ CategoryTheory.forget₂ SemiRingCatMax MonCat) ⋙ CategoryTheory.forget MonCat).obj j)

full_name: SemiRingCat.FilteredColimits.R
code: /-- The colimit of `F ⋙ forget₂ SemiRingCat MonCat` in the category `MonCat`.
In the following, we will show that this has the structure of a semiring.
-/
abbrev R : MonCatMax.{v, u} 
goal: MonCatMax

full_name: SemiRingCat.FilteredColimits.colimitSemiring
code: instance colimitSemiring : Semiring.{max v u} <| R.{v, u} F 
goal: Semiring ↑(SemiRingCat.FilteredColimits.R F)

full_name: SemiRingCat.FilteredColimits.colimit
code: /-- The bundled semiring giving the filtered colimit of a diagram. -/
def colimit : SemiRingCatMax.{v, u} 
goal: SemiRingCatMax

full_name: SemiRingCat.FilteredColimits.colimitCocone
code: /-- The cocone over the proposed colimit semiring. -/
def colimitCocone : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: SemiRingCat.FilteredColimits.colimitCoconeIsColimit
code: /-- The proposed colimit cocone is a colimit in `SemiRingCat`. -/
def colimitCoconeIsColimit : IsColimit <| colimitCocone.{v, u} F 
goal: CategoryTheory.Limits.IsColimit (SemiRingCat.FilteredColimits.colimitCocone F)

full_name: SemiRingCat.FilteredColimits.forget₂MonPreservesFilteredColimits
code: instance forget₂MonPreservesFilteredColimits :
    PreservesFilteredColimits (forget₂ SemiRingCat MonCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget₂ SemiRingCat MonCat)

full_name: SemiRingCat.FilteredColimits.forgetPreservesFilteredColimits
code: instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget SemiRingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget SemiRingCat)

full_name: CommSemiRingCat.FilteredColimits.R
code: /-- The colimit of `F ⋙ forget₂ CommSemiRingCat SemiRingCat` in the category `SemiRingCat`.
In the following, we will show that this has the structure of a _commutative_ semiring.
-/
abbrev R : SemiRingCatMax.{v, u} 
goal: SemiRingCatMax

full_name: CommSemiRingCat.FilteredColimits.colimitCommSemiring
code: instance colimitCommSemiring : CommSemiring.{max v u} <| R.{v, u} F 
goal: CommSemiring ↑(CommSemiRingCat.FilteredColimits.R F)

full_name: CommSemiRingCat.FilteredColimits.colimit
code: /-- The bundled commutative semiring giving the filtered colimit of a diagram. -/
def colimit : CommSemiRingCat.{max v u} 
goal: CommSemiRingCat

full_name: CommSemiRingCat.FilteredColimits.colimitCocone
code: /-- The cocone over the proposed colimit commutative semiring. -/
def colimitCocone : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: CommSemiRingCat.FilteredColimits.colimitCoconeIsColimit
code: /-- The proposed colimit cocone is a colimit in `CommSemiRingCat`. -/
def colimitCoconeIsColimit : IsColimit <| colimitCocone.{v, u} F 
goal: CategoryTheory.Limits.IsColimit (CommSemiRingCat.FilteredColimits.colimitCocone F)

full_name: CommSemiRingCat.FilteredColimits.forget₂SemiRingPreservesFilteredColimits
code: instance forget₂SemiRingPreservesFilteredColimits :
    PreservesFilteredColimits (forget₂ CommSemiRingCat SemiRingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget₂ CommSemiRingCat SemiRingCat)

full_name: CommSemiRingCat.FilteredColimits.forgetPreservesFilteredColimits
code: instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget CommSemiRingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget CommSemiRingCat)

full_name: RingCat.FilteredColimits.R
code: /-- The colimit of `F ⋙ forget₂ RingCat SemiRingCat` in the category `SemiRingCat`.
In the following, we will show that this has the structure of a ring.
-/
abbrev R : SemiRingCat.{max v u} 
goal: SemiRingCat

full_name: RingCat.FilteredColimits.colimitRing
code: instance colimitRing : Ring.{max v u} <| R.{v, u} F 
goal: Ring ↑(RingCat.FilteredColimits.R F)

full_name: RingCat.FilteredColimits.colimit
code: /-- The bundled ring giving the filtered colimit of a diagram. -/
def colimit : RingCat.{max v u} 
goal: RingCat

full_name: RingCat.FilteredColimits.colimitCocone
code: /-- The cocone over the proposed colimit ring. -/
def colimitCocone : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: RingCat.FilteredColimits.colimitCoconeIsColimit
code: /-- The proposed colimit cocone is a colimit in `Ring`. -/
def colimitCoconeIsColimit : IsColimit <| colimitCocone.{v, u} F 
goal: CategoryTheory.Limits.IsColimit (RingCat.FilteredColimits.colimitCocone F)

full_name: RingCat.FilteredColimits.forget₂SemiRingPreservesFilteredColimits
code: instance forget₂SemiRingPreservesFilteredColimits :
    PreservesFilteredColimits (forget₂ RingCat SemiRingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget₂ RingCat SemiRingCat)

full_name: RingCat.FilteredColimits.forgetPreservesFilteredColimits
code: instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget RingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget RingCat)

full_name: CommRingCat.FilteredColimits.R
code: /-- The colimit of `F ⋙ forget₂ CommRingCat RingCat` in the category `RingCat`.
In the following, we will show that this has the structure of a _commutative_ ring.
-/
abbrev R : RingCat.{max v u} 
goal: RingCat

full_name: CommRingCat.FilteredColimits.colimitCommRing
code: instance colimitCommRing : CommRing.{max v u} <| R.{v, u} F 
goal: CommRing ↑(CommRingCat.FilteredColimits.R F)

full_name: CommRingCat.FilteredColimits.colimit
code: /-- The bundled commutative ring giving the filtered colimit of a diagram. -/
def colimit : CommRingCat.{max v u} 
goal: CommRingCat

full_name: CommRingCat.FilteredColimits.colimitCocone
code: /-- The cocone over the proposed colimit commutative ring. -/
def colimitCocone : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: CommRingCat.FilteredColimits.colimitCoconeIsColimit
code: /-- The proposed colimit cocone is a colimit in `CommRingCat`. -/
def colimitCoconeIsColimit : IsColimit <| colimitCocone.{v, u} F 
goal: CategoryTheory.Limits.IsColimit (CommRingCat.FilteredColimits.colimitCocone F)

full_name: CommRingCat.FilteredColimits.forget₂RingPreservesFilteredColimits
code: instance forget₂RingPreservesFilteredColimits :
    PreservesFilteredColimits (forget₂ CommRingCat RingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget₂ CommRingCat RingCat)

full_name: CommRingCat.FilteredColimits.forgetPreservesFilteredColimits
code: instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget CommRingCat.{u}) 
goal: CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget CommRingCat)