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/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limit preservation properties of `Functor.op` and related constructions
We formulate conditions about `F` which imply that `F.op`, `F.unop`, `F.leftOp` and `F.rightOp`
preserve certain (co)limits and vice versa.
-/
universe w w' v₁ v₂ u₁ u₂
noncomputable section
open CategoryTheory
namespace CategoryTheory.Limits
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
variable {J : Type w} [Category.{w'} J]
/-- If `F : C ⥤ D` preserves colimits of `K.leftOp : Jᵒᵖ ⥤ C`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves
limits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesLimit_op (K : J ⥤ Cᵒᵖ) (F : C ⥤ D) [PreservesColimit K.leftOp F] :
PreservesLimit K F.op where
preserves {_} hc :=
⟨isLimitConeRightOpOfCocone _ (isColimitOfPreserves F (isColimitCoconeLeftOpOfCone _ hc))⟩
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F : C ⥤ D` preserves
limits of `K : J ⥤ C`. -/
lemma preservesLimit_of_op (K : J ⥤ C) (F : C ⥤ D) [PreservesColimit K.op F.op] :
PreservesLimit K F where
preserves {_} hc := ⟨isLimitOfOp (isColimitOfPreserves F.op (IsLimit.op hc))⟩
/-- If `F : C ⥤ Dᵒᵖ` preserves colimits of `K.leftOp : Jᵒᵖ ⥤ C`, then `F.leftOp : Cᵒᵖ ⥤ D`
preserves limits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesLimit_leftOp (K : J ⥤ Cᵒᵖ) (F : C ⥤ Dᵒᵖ) [PreservesColimit K.leftOp F] :
PreservesLimit K F.leftOp where
preserves {_} hc :=
⟨isLimitConeUnopOfCocone _ (isColimitOfPreserves F (isColimitCoconeLeftOpOfCone _ hc))⟩
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F : C ⥤ Dᵒᵖ` preserves
limits of `K : J ⥤ C`. -/
lemma preservesLimit_of_leftOp (K : J ⥤ C) (F : C ⥤ Dᵒᵖ) [PreservesColimit K.op F.leftOp] :
PreservesLimit K F where
preserves {_} hc :=
⟨isLimitOfCoconeLeftOpOfCone _ (isColimitOfPreserves F.leftOp (IsLimit.op hc))⟩
/-- If `F : Cᵒᵖ ⥤ D` preserves colimits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.rightOp : C ⥤ Dᵒᵖ` preserves
limits of `K : J ⥤ C`. -/
lemma preservesLimit_rightOp (K : J ⥤ C) (F : Cᵒᵖ ⥤ D) [PreservesColimit K.op F] :
PreservesLimit K F.rightOp where
preserves {_} hc :=
⟨isLimitConeRightOpOfCocone _ (isColimitOfPreserves F hc.op)⟩
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits of `K.leftOp : Jᵒᵖ ⥤ Cᵒᵖ`, then `F : Cᵒᵖ ⥤ D`
preserves limits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesLimit_of_rightOp (K : J ⥤ Cᵒᵖ) (F : Cᵒᵖ ⥤ D) [PreservesColimit K.leftOp F.rightOp] :
PreservesLimit K F where
preserves {_} hc :=
⟨isLimitOfOp (isColimitOfPreserves F.rightOp (isColimitCoconeLeftOpOfCone _ hc))⟩
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.unop : C ⥤ D` preserves
limits of `K : J ⥤ C`. -/
lemma preservesLimit_unop (K : J ⥤ C) (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimit K.op F] :
PreservesLimit K F.unop where
preserves {_} hc :=
⟨isLimitConeUnopOfCocone _ (isColimitOfPreserves F hc.op)⟩
/-- If `F.unop : C ⥤ D` preserves colimits of `K.leftOp : Jᵒᵖ ⥤ C`, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves
limits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesLimit_of_unop (K : J ⥤ Cᵒᵖ) (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimit K.leftOp F.unop] :
PreservesLimit K F where
preserves {_} hc :=
⟨isLimitOfCoconeLeftOpOfCone _ (isColimitOfPreserves F.unop (isColimitCoconeLeftOpOfCone _ hc))⟩
/-- If `F : C ⥤ D` preserves limits of `K.leftOp : Jᵒᵖ ⥤ C`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves
colimits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesColimit_op (K : J ⥤ Cᵒᵖ) (F : C ⥤ D) [PreservesLimit K.leftOp F] :
PreservesColimit K F.op where
preserves {_} hc :=
⟨isColimitCoconeRightOpOfCone _ (isLimitOfPreserves F (isLimitConeLeftOpOfCocone _ hc))⟩
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F : C ⥤ D` preserves
colimits of `K : J ⥤ C`. -/
lemma preservesColimit_of_op (K : J ⥤ C) (F : C ⥤ D) [PreservesLimit K.op F.op] :
PreservesColimit K F where
preserves {_} hc := ⟨isColimitOfOp (isLimitOfPreserves F.op (IsColimit.op hc))⟩
/-- If `F : C ⥤ Dᵒᵖ` preserves limits of `K.leftOp : Jᵒᵖ ⥤ C`, then `F.leftOp : Cᵒᵖ ⥤ D` preserves
colimits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesColimit_leftOp (K : J ⥤ Cᵒᵖ) (F : C ⥤ Dᵒᵖ) [PreservesLimit K.leftOp F] :
PreservesColimit K F.leftOp where
preserves {_} hc :=
⟨isColimitCoconeUnopOfCone _ (isLimitOfPreserves F (isLimitConeLeftOpOfCocone _ hc))⟩
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves limits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F : C ⥤ Dᵒᵖ` preserves
colimits of `K : J ⥤ C`. -/
lemma preservesColimit_of_leftOp (K : J ⥤ C) (F : C ⥤ Dᵒᵖ) [PreservesLimit K.op F.leftOp] :
PreservesColimit K F where
preserves {_} hc :=
⟨isColimitOfConeLeftOpOfCocone _ (isLimitOfPreserves F.leftOp (IsColimit.op hc))⟩
/-- If `F : Cᵒᵖ ⥤ D` preserves limits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.rightOp : C ⥤ Dᵒᵖ` preserves
colimits of `K : J ⥤ C`. -/
lemma preservesColimit_rightOp (K : J ⥤ C) (F : Cᵒᵖ ⥤ D) [PreservesLimit K.op F] :
PreservesColimit K F.rightOp where
preserves {_} hc :=
⟨isColimitCoconeRightOpOfCone _ (isLimitOfPreserves F hc.op)⟩
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves limits of `K.leftOp : Jᵒᵖ ⥤ C`, then `F : Cᵒᵖ ⥤ D`
preserves colimits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesColimit_of_rightOp (K : J ⥤ Cᵒᵖ) (F : Cᵒᵖ ⥤ D) [PreservesLimit K.leftOp F.rightOp] :
PreservesColimit K F where
preserves {_} hc :=
⟨isColimitOfOp (isLimitOfPreserves F.rightOp (isLimitConeLeftOpOfCocone _ hc))⟩
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.unop : C ⥤ D` preserves
colimits of `K : J ⥤ C`. -/
lemma preservesColimit_unop (K : J ⥤ C) (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimit K.op F] :
PreservesColimit K F.unop where
preserves {_} hc :=
⟨isColimitCoconeUnopOfCone _ (isLimitOfPreserves F hc.op)⟩
/-- If `F.unop : C ⥤ D` preserves limits of `K.op : Jᵒᵖ ⥤ C`, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves
colimits of `K : J ⥤ Cᵒᵖ`. -/
lemma preservesColimit_of_unop (K : J ⥤ Cᵒᵖ) (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimit K.leftOp F.unop] :
PreservesColimit K F where
preserves {_} hc :=
⟨isColimitOfConeLeftOpOfCocone _ (isLimitOfPreserves F.unop (isLimitConeLeftOpOfCocone _ hc))⟩
section
variable (J)
/-- If `F : C ⥤ D` preserves colimits of shape `Jᵒᵖ`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of
shape `J`. -/
lemma preservesLimitsOfShape_op (F : C ⥤ D) [PreservesColimitsOfShape Jᵒᵖ F] :
PreservesLimitsOfShape J F.op where preservesLimit {K} := preservesLimit_op K F
/-- If `F : C ⥤ Dᵒᵖ` preserves colimits of shape `Jᵒᵖ`, then `F.leftOp : Cᵒᵖ ⥤ D` preserves limits
of shape `J`. -/
lemma preservesLimitsOfShape_leftOp (F : C ⥤ Dᵒᵖ) [PreservesColimitsOfShape Jᵒᵖ F] :
PreservesLimitsOfShape J F.leftOp where preservesLimit {K} := preservesLimit_leftOp K F
/-- If `F : Cᵒᵖ ⥤ D` preserves colimits of shape `Jᵒᵖ`, then `F.rightOp : C ⥤ Dᵒᵖ` preserves limits
of shape `J`. -/
lemma preservesLimitsOfShape_rightOp (F : Cᵒᵖ ⥤ D) [PreservesColimitsOfShape Jᵒᵖ F] :
PreservesLimitsOfShape J F.rightOp where preservesLimit {K} := preservesLimit_rightOp K F
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of shape `Jᵒᵖ`, then `F.unop : C ⥤ D` preserves limits of
shape `J`. -/
lemma preservesLimitsOfShape_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimitsOfShape Jᵒᵖ F] :
PreservesLimitsOfShape J F.unop where preservesLimit {K} := preservesLimit_unop K F
/-- If `F : C ⥤ D` preserves limits of shape `Jᵒᵖ`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of
shape `J`. -/
lemma preservesColimitsOfShape_op (F : C ⥤ D) [PreservesLimitsOfShape Jᵒᵖ F] :
PreservesColimitsOfShape J F.op where preservesColimit {K} := preservesColimit_op K F
/-- If `F : C ⥤ Dᵒᵖ` preserves limits of shape `Jᵒᵖ`, then `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_leftOp (F : C ⥤ Dᵒᵖ) [PreservesLimitsOfShape Jᵒᵖ F] :
PreservesColimitsOfShape J F.leftOp where preservesColimit {K} := preservesColimit_leftOp K F
/-- If `F : Cᵒᵖ ⥤ D` preserves limits of shape `Jᵒᵖ`, then `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_rightOp (F : Cᵒᵖ ⥤ D) [PreservesLimitsOfShape Jᵒᵖ F] :
PreservesColimitsOfShape J F.rightOp where preservesColimit {K} := preservesColimit_rightOp K F
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of shape `Jᵒᵖ`, then `F.unop : C ⥤ D` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimitsOfShape Jᵒᵖ F] :
PreservesColimitsOfShape J F.unop where preservesColimit {K} := preservesColimit_unop K F
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of shape `Jᵒᵖ`, then `F : C ⥤ D` preserves limits
of shape `J`. -/
lemma preservesLimitsOfShape_of_op (F : C ⥤ D) [PreservesColimitsOfShape Jᵒᵖ F.op] :
PreservesLimitsOfShape J F where preservesLimit {K} := preservesLimit_of_op K F
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits of shape `Jᵒᵖ`, then `F : C ⥤ Dᵒᵖ` preserves limits
of shape `J`. -/
lemma preservesLimitsOfShape_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesColimitsOfShape Jᵒᵖ F.leftOp] :
PreservesLimitsOfShape J F where preservesLimit {K} := preservesLimit_of_leftOp K F
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits of shape `Jᵒᵖ`, then `F : Cᵒᵖ ⥤ D` preserves limits
of shape `J`. -/
lemma preservesLimitsOfShape_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesColimitsOfShape Jᵒᵖ F.rightOp] :
PreservesLimitsOfShape J F where preservesLimit {K} := preservesLimit_of_rightOp K F
/-- If `F.unop : C ⥤ D` preserves colimits of shape `Jᵒᵖ`, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits
of shape `J`. -/
lemma preservesLimitsOfShape_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimitsOfShape Jᵒᵖ F.unop] :
PreservesLimitsOfShape J F where preservesLimit {K} := preservesLimit_of_unop K F
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of shape `Jᵒᵖ`, then `F : C ⥤ D` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_of_op (F : C ⥤ D) [PreservesLimitsOfShape Jᵒᵖ F.op] :
PreservesColimitsOfShape J F where preservesColimit {K} := preservesColimit_of_op K F
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves limits of shape `Jᵒᵖ`, then `F : C ⥤ Dᵒᵖ` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesLimitsOfShape Jᵒᵖ F.leftOp] :
PreservesColimitsOfShape J F where preservesColimit {K} := preservesColimit_of_leftOp K F
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves limits of shape `Jᵒᵖ`, then `F : Cᵒᵖ ⥤ D` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesLimitsOfShape Jᵒᵖ F.rightOp] :
PreservesColimitsOfShape J F where preservesColimit {K} := preservesColimit_of_rightOp K F
/-- If `F.unop : C ⥤ D` preserves limits of shape `Jᵒᵖ`, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits
of shape `J`. -/
lemma preservesColimitsOfShape_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimitsOfShape Jᵒᵖ F.unop] :
PreservesColimitsOfShape J F where preservesColimit {K} := preservesColimit_of_unop K F
end
/-- If `F : C ⥤ D` preserves colimits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimitsOfSize_op (F : C ⥤ D) [PreservesColimitsOfSize.{w, w'} F] :
PreservesLimitsOfSize.{w, w'} F.op where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_op _ _
/-- If `F : C ⥤ Dᵒᵖ` preserves colimits, then `F.leftOp : Cᵒᵖ ⥤ D` preserves limits. -/
lemma preservesLimitsOfSize_leftOp (F : C ⥤ Dᵒᵖ) [PreservesColimitsOfSize.{w, w'} F] :
PreservesLimitsOfSize.{w, w'} F.leftOp where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_leftOp _ _
/-- If `F : Cᵒᵖ ⥤ D` preserves colimits, then `F.rightOp : C ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimitsOfSize_rightOp (F : Cᵒᵖ ⥤ D) [PreservesColimitsOfSize.{w, w'} F] :
PreservesLimitsOfSize.{w, w'} F.rightOp where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_rightOp _ _
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits, then `F.unop : C ⥤ D` preserves limits. -/
lemma preservesLimitsOfSize_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimitsOfSize.{w, w'} F] :
PreservesLimitsOfSize.{w, w'} F.unop where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_unop _ _
/-- If `F : C ⥤ D` preserves limits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimitsOfSize_op (F : C ⥤ D) [PreservesLimitsOfSize.{w, w'} F] :
PreservesColimitsOfSize.{w, w'} F.op where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_op _ _
/-- If `F : C ⥤ Dᵒᵖ` preserves limits, then `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits. -/
lemma preservesColimitsOfSize_leftOp (F : C ⥤ Dᵒᵖ) [PreservesLimitsOfSize.{w, w'} F] :
PreservesColimitsOfSize.{w, w'} F.leftOp where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_leftOp _ _
/-- If `F : Cᵒᵖ ⥤ D` preserves limits, then `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimitsOfSize_rightOp (F : Cᵒᵖ ⥤ D) [PreservesLimitsOfSize.{w, w'} F] :
PreservesColimitsOfSize.{w, w'} F.rightOp where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_rightOp _ _
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits, then `F.unop : C ⥤ D` preserves colimits. -/
lemma preservesColimitsOfSize_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimitsOfSize.{w, w'} F] :
PreservesColimitsOfSize.{w, w'} F.unop where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_unop _ _
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits, then `F : C ⥤ D` preserves limits. -/
lemma preservesLimitsOfSize_of_op (F : C ⥤ D) [PreservesColimitsOfSize.{w, w'} F.op] :
PreservesLimitsOfSize.{w, w'} F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_op _ _
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits, then `F : C ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimitsOfSize_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesColimitsOfSize.{w, w'} F.leftOp] :
PreservesLimitsOfSize.{w, w'} F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_leftOp _ _
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits, then `F : Cᵒᵖ ⥤ D` preserves limits. -/
lemma preservesLimitsOfSize_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesColimitsOfSize.{w, w'} F.rightOp] :
PreservesLimitsOfSize.{w, w'} F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_rightOp _ _
/-- If `F.unop : C ⥤ D` preserves colimits, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimitsOfSize_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimitsOfSize.{w, w'} F.unop] :
PreservesLimitsOfSize.{w, w'} F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_unop _ _
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits, then `F : C ⥤ D` preserves colimits. -/
lemma preservesColimitsOfSize_of_op (F : C ⥤ D) [PreservesLimitsOfSize.{w, w'} F.op] :
PreservesColimitsOfSize.{w, w'} F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_op _ _
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves limits, then `F : C ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimitsOfSize_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesLimitsOfSize.{w, w'} F.leftOp] :
PreservesColimitsOfSize.{w, w'} F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_leftOp _ _
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves limits, then `F : Cᵒᵖ ⥤ D` preserves colimits. -/
lemma preservesColimitsOfSize_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesLimitsOfSize.{w, w'} F.rightOp] :
PreservesColimitsOfSize.{w, w'} F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_rightOp _ _
/-- If `F.unop : C ⥤ D` preserves limits, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimitsOfSize_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimitsOfSize.{w, w'} F.unop] :
PreservesColimitsOfSize.{w, w'} F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_unop _ _
/-- If `F : C ⥤ D` preserves colimits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimits_op (F : C ⥤ D) [PreservesColimits F] : PreservesLimits F.op where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_op _ _
/-- If `F : C ⥤ Dᵒᵖ` preserves colimits, then `F.leftOp : Cᵒᵖ ⥤ D` preserves limits. -/
lemma preservesLimits_leftOp (F : C ⥤ Dᵒᵖ) [PreservesColimits F] : PreservesLimits F.leftOp where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_leftOp _ _
/-- If `F : Cᵒᵖ ⥤ D` preserves colimits, then `F.rightOp : C ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimits_rightOp (F : Cᵒᵖ ⥤ D) [PreservesColimits F] : PreservesLimits F.rightOp where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_rightOp _ _
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits, then `F.unop : C ⥤ D` preserves limits. -/
lemma preservesLimits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimits F] : PreservesLimits F.unop where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_unop _ _
/-- If `F : C ⥤ D` preserves limits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimits_op (F : C ⥤ D) [PreservesLimits F] : PreservesColimits F.op where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_op _ _
@[deprecated (since := "2024-12-25")] alias perservesColimits_op := preservesColimits_op
/-- If `F : C ⥤ Dᵒᵖ` preserves limits, then `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits. -/
lemma preservesColimits_leftOp (F : C ⥤ Dᵒᵖ) [PreservesLimits F] : PreservesColimits F.leftOp where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_leftOp _ _
/-- If `F : Cᵒᵖ ⥤ D` preserves limits, then `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimits_rightOp (F : Cᵒᵖ ⥤ D) [PreservesLimits F] :
PreservesColimits F.rightOp where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_rightOp _ _
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits, then `F.unop : C ⥤ D` preserves colimits. -/
lemma preservesColimits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimits F] : PreservesColimits F.unop where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_unop _ _
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits, then `F : C ⥤ D` preserves limits. -/
lemma preservesLimits_of_op (F : C ⥤ D) [PreservesColimits F.op] : PreservesLimits F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_op _ _
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves colimits, then `F : C ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimits_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesColimits F.leftOp] : PreservesLimits F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_leftOp _ _
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves colimits, then `F : Cᵒᵖ ⥤ D` preserves limits. -/
lemma preservesLimits_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesColimits F.rightOp] :
PreservesLimits F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_rightOp _ _
/-- If `F.unop : C ⥤ D` preserves colimits, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits. -/
lemma preservesLimits_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesColimits F.unop] : PreservesLimits F where
preservesLimitsOfShape {_} _ := preservesLimitsOfShape_of_unop _ _
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits, then `F : C ⥤ D` preserves colimits. -/
lemma preservesColimits_of_op (F : C ⥤ D) [PreservesLimits F.op] : PreservesColimits F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_op _ _
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves limits, then `F : C ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimits_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesLimits F.leftOp] :
PreservesColimits F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_leftOp _ _
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves limits, then `F : Cᵒᵖ ⥤ D` preserves colimits. -/
lemma preservesColimits_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesLimits F.rightOp] :
PreservesColimits F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_rightOp _ _
/-- If `F.unop : C ⥤ D` preserves limits, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits. -/
lemma preservesColimits_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesLimits F.unop] : PreservesColimits F where
preservesColimitsOfShape {_} _ := preservesColimitsOfShape_of_unop _ _
/-- If `F : C ⥤ D` preserves finite colimits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite
limits. -/
lemma preservesFiniteLimits_op (F : C ⥤ D) [PreservesFiniteColimits F] :
PreservesFiniteLimits F.op where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_op J F
/-- If `F : C ⥤ Dᵒᵖ` preserves finite colimits, then `F.leftOp : Cᵒᵖ ⥤ D` preserves finite
limits. -/
lemma preservesFiniteLimits_leftOp (F : C ⥤ Dᵒᵖ) [PreservesFiniteColimits F] :
PreservesFiniteLimits F.leftOp where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_leftOp J F
/-- If `F : Cᵒᵖ ⥤ D` preserves finite colimits, then `F.rightOp : C ⥤ Dᵒᵖ` preserves finite
limits. -/
lemma preservesFiniteLimits_rightOp (F : Cᵒᵖ ⥤ D) [PreservesFiniteColimits F] :
PreservesFiniteLimits F.rightOp where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_rightOp J F
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite colimits, then `F.unop : C ⥤ D` preserves finite
limits. -/
lemma preservesFiniteLimits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesFiniteColimits F] :
PreservesFiniteLimits F.unop where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_unop J F
/-- If `F : C ⥤ D` preserves finite limits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite
colimits. -/
lemma preservesFiniteColimits_op (F : C ⥤ D) [PreservesFiniteLimits F] :
PreservesFiniteColimits F.op where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_op J F
/-- If `F : C ⥤ Dᵒᵖ` preserves finite limits, then `F.leftOp : Cᵒᵖ ⥤ D` preserves finite
colimits. -/
lemma preservesFiniteColimits_leftOp (F : C ⥤ Dᵒᵖ) [PreservesFiniteLimits F] :
PreservesFiniteColimits F.leftOp where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_leftOp J F
/-- If `F : Cᵒᵖ ⥤ D` preserves finite limits, then `F.rightOp : C ⥤ Dᵒᵖ` preserves finite
colimits. -/
lemma preservesFiniteColimits_rightOp (F : Cᵒᵖ ⥤ D) [PreservesFiniteLimits F] :
PreservesFiniteColimits F.rightOp where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_rightOp J F
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite limits, then `F.unop : C ⥤ D` preserves finite
colimits. -/
lemma preservesFiniteColimits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesFiniteLimits F] :
PreservesFiniteColimits F.unop where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_unop J F
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite colimits, then `F : C ⥤ D` preserves finite limits. -/
lemma preservesFiniteLimits_of_op (F : C ⥤ D) [PreservesFiniteColimits F.op] :
PreservesFiniteLimits F where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_of_op J F
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves finite colimits, then `F : C ⥤ Dᵒᵖ` preserves finite
limits. -/
lemma preservesFiniteLimits_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesFiniteColimits F.leftOp] :
PreservesFiniteLimits F where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_of_leftOp J F
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves finite colimits, then `F : Cᵒᵖ ⥤ D` preserves finite
limits. -/
lemma preservesFiniteLimits_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesFiniteColimits F.rightOp] :
PreservesFiniteLimits F where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_of_rightOp J F
/-- If `F.unop : C ⥤ D` preserves finite colimits, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite limits. -/
lemma preservesFiniteLimits_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesFiniteColimits F.unop] :
PreservesFiniteLimits F where
preservesFiniteLimits J _ _ := preservesLimitsOfShape_of_unop J F
/-- If `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite limits, then `F : C ⥤ D` preserves finite colimits. -/
lemma preservesFiniteColimits_of_op (F : C ⥤ D) [PreservesFiniteLimits F.op] :
PreservesFiniteColimits F where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_of_op J F
/-- If `F.leftOp : Cᵒᵖ ⥤ D` preserves finite limits, then `F : C ⥤ Dᵒᵖ` preserves finite
colimits. -/
lemma preservesFiniteColimits_of_leftOp (F : C ⥤ Dᵒᵖ) [PreservesFiniteLimits F.leftOp] :
PreservesFiniteColimits F where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_of_leftOp J F
/-- If `F.rightOp : C ⥤ Dᵒᵖ` preserves finite limits, then `F : Cᵒᵖ ⥤ D` preserves finite
colimits. -/
lemma preservesFiniteColimits_of_rightOp (F : Cᵒᵖ ⥤ D) [PreservesFiniteLimits F.rightOp] :
PreservesFiniteColimits F where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_of_rightOp J F
/-- If `F.unop : C ⥤ D` preserves finite limits, then `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite colimits. -/
lemma preservesFiniteColimits_of_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesFiniteLimits F.unop] :
PreservesFiniteColimits F where
preservesFiniteColimits J _ _ := preservesColimitsOfShape_of_unop J F
/-- If `F : C ⥤ D` preserves finite coproducts, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite
products. -/
lemma preservesFiniteProducts_op (F : C ⥤ D) [PreservesFiniteCoproducts F] :
PreservesFiniteProducts F.op where
preserves n := by
apply (config := { allowSynthFailures := true }) preservesLimitsOfShape_op
exact preservesColimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : C ⥤ Dᵒᵖ` preserves finite coproducts, then `F.leftOp : Cᵒᵖ ⥤ D` preserves finite
products. -/
lemma preservesFiniteProducts_leftOp (F : C ⥤ Dᵒᵖ) [PreservesFiniteCoproducts F] :
PreservesFiniteProducts F.leftOp where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesLimitsOfShape_leftOp
exact preservesColimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : Cᵒᵖ ⥤ D` preserves finite coproducts, then `F.rightOp : C ⥤ Dᵒᵖ` preserves finite
products. -/
lemma preservesFiniteProducts_rightOp (F : Cᵒᵖ ⥤ D) [PreservesFiniteCoproducts F] :
PreservesFiniteProducts F.rightOp where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesLimitsOfShape_rightOp
exact preservesColimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite coproducts, then `F.unop : C ⥤ D` preserves finite
products. -/
lemma preservesFiniteProducts_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesFiniteCoproducts F] :
PreservesFiniteProducts F.unop where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesLimitsOfShape_unop
exact preservesColimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : C ⥤ D` preserves finite products, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite
coproducts. -/
lemma preservesFiniteCoproducts_op (F : C ⥤ D) [PreservesFiniteProducts F] :
PreservesFiniteCoproducts F.op where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesColimitsOfShape_op
exact preservesLimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : C ⥤ Dᵒᵖ` preserves finite products, then `F.leftOp : Cᵒᵖ ⥤ D` preserves finite
coproducts. -/
lemma preservesFiniteCoproducts_leftOp (F : C ⥤ Dᵒᵖ) [PreservesFiniteProducts F] :
PreservesFiniteCoproducts F.leftOp where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesColimitsOfShape_leftOp
exact preservesLimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : Cᵒᵖ ⥤ D` preserves finite products, then `F.rightOp : C ⥤ Dᵒᵖ` preserves finite
coproducts. -/
lemma preservesFiniteCoproducts_rightOp (F : Cᵒᵖ ⥤ D) [PreservesFiniteProducts F] :
PreservesFiniteCoproducts F.rightOp where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesColimitsOfShape_rightOp
exact preservesLimitsOfShape_of_equiv (Discrete.opposite _).symm _
/-- If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite products, then `F.unop : C ⥤ D` preserves finite
coproducts. -/
lemma preservesFiniteCoproducts_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [PreservesFiniteProducts F] :
PreservesFiniteCoproducts F.unop where
preserves _ := by
apply (config := { allowSynthFailures := true }) preservesColimitsOfShape_unop
exact preservesLimitsOfShape_of_equiv (Discrete.opposite _).symm _
end CategoryTheory.Limits