Categories with finite limits.

A typeclass for categories with all finite (co)limits.

class CategoryTheory.Limits.HasFiniteLimits (C : Type u) [CategoryTheory.Category.{v, u} C] : Prop

• out : ∀ (J : Type) [𝒥 : CategoryTheory.SmallCategory J] [inst : CategoryTheory.FinCategory J], CategoryTheory.Limits.HasLimitsOfShape J C

  C has all limits over any type J whose objects and morphisms lie in the same universe and which has FinType objects and morphisms

A category has all finite limits if every functor J ⥤ C with a FinCategory J instance and J : Type has a limit.

This is often called 'finitely complete'.

instance CategoryTheory.Limits.hasLimitsOfShape_of_hasFiniteLimits (C : Type u) [CategoryTheory.Category.{v, u} C] (J : Type w) [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] [CategoryTheory.Limits.HasFiniteLimits C] : CategoryTheory.Limits.HasLimitsOfShape J C

Equations

• CategoryTheory.Limits.hasLimitsOfShape_of_hasFiniteLimits = CategoryTheory.Limits.hasLimitsOfShape_of_hasFiniteLimits.proof_1

instance CategoryTheory.Limits.hasFiniteLimits_of_hasLimitsOfSize (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfSize.{v', u', v, u} C] : CategoryTheory.Limits.HasFiniteLimits C

Equations

• CategoryTheory.Limits.hasFiniteLimits_of_hasLimitsOfSize = CategoryTheory.Limits.hasFiniteLimits_of_hasLimitsOfSize.proof_1

instance CategoryTheory.Limits.hasFiniteLimits_of_hasLimits (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C] : CategoryTheory.Limits.HasFiniteLimits C

If C has all limits, it has finite limits.

Equations

• CategoryTheory.Limits.hasFiniteLimits_of_hasLimits = CategoryTheory.Limits.hasFiniteLimits_of_hasLimits.proof_1

theorem CategoryTheory.Limits.hasFiniteLimits_of_hasFiniteLimits_of_size (C : Type u) [CategoryTheory.Category.{v, u} C] (h : ∀ (J : Type w) {𝒥 : CategoryTheory.SmallCategory J}, CategoryTheory.FinCategory J → CategoryTheory.Limits.HasLimitsOfShape J C) : CategoryTheory.Limits.HasFiniteLimits C

We can always derive HasFiniteLimits C by providing limits at an arbitrary universe.

class CategoryTheory.Limits.HasFiniteColimits (C : Type u) [CategoryTheory.Category.{v, u} C] : Prop

• out : ∀ (J : Type) [𝒥 : CategoryTheory.SmallCategory J] [inst : CategoryTheory.FinCategory J], CategoryTheory.Limits.HasColimitsOfShape J C

  C has all colimits over any type J whose objects and morphisms lie in the same universe and which has Fintype objects and morphisms

A category has all finite colimits if every functor J ⥤ C with a FinCategory J instance and J : Type has a colimit.

This is often called 'finitely cocomplete'.

instance CategoryTheory.Limits.hasColimitsOfShape_of_hasFiniteColimits (C : Type u) [CategoryTheory.Category.{v, u} C] (J : Type w) [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] [CategoryTheory.Limits.HasFiniteColimits C] : CategoryTheory.Limits.HasColimitsOfShape J C

Equations

• CategoryTheory.Limits.hasColimitsOfShape_of_hasFiniteColimits = CategoryTheory.Limits.hasColimitsOfShape_of_hasFiniteColimits.proof_1

instance CategoryTheory.Limits.hasFiniteColimits_of_hasColimitsOfSize (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimitsOfSize.{v', u', v, u} C] : CategoryTheory.Limits.HasFiniteColimits C

Equations

• CategoryTheory.Limits.hasFiniteColimits_of_hasColimitsOfSize = CategoryTheory.Limits.hasFiniteColimits_of_hasColimitsOfSize.proof_1

theorem CategoryTheory.Limits.hasFiniteColimits_of_hasFiniteColimits_of_size (C : Type u) [CategoryTheory.Category.{v, u} C] (h : ∀ (J : Type w) {𝒥 : CategoryTheory.SmallCategory J}, CategoryTheory.FinCategory J → CategoryTheory.Limits.HasColimitsOfShape J C) : CategoryTheory.Limits.HasFiniteColimits C

We can always derive HasFiniteColimits C by providing colimits at an arbitrary universe.

instance CategoryTheory.Limits.fintypeWalkingParallelPair : Fintype CategoryTheory.Limits.WalkingParallelPair

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• One or more equations did not get rendered due to their size.

instance CategoryTheory.Limits.instFintypeWalkingParallelPairHom (j : CategoryTheory.Limits.WalkingParallelPair) (j' : CategoryTheory.Limits.WalkingParallelPair) : Fintype (CategoryTheory.Limits.WalkingParallelPairHom j j')

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• One or more equations did not get rendered due to their size.

instance CategoryTheory.Limits.instFinCategoryWalkingParallelPairWalkingParallelPairHomCategory : CategoryTheory.FinCategory CategoryTheory.Limits.WalkingParallelPair

Equations

• CategoryTheory.Limits.instFinCategoryWalkingParallelPairWalkingParallelPairHomCategory = CategoryTheory.FinCategory.mk

instance CategoryTheory.Limits.WidePullbackShape.fintypeObj {J : Type v} [Fintype J] : Fintype (CategoryTheory.Limits.WidePullbackShape J)

Equations

• CategoryTheory.Limits.WidePullbackShape.fintypeObj = Eq.mpr (_ : Fintype (CategoryTheory.Limits.WidePullbackShape J) = Fintype (Option J)) inferInstance

instance CategoryTheory.Limits.WidePullbackShape.fintypeHom {J : Type v} (j : CategoryTheory.Limits.WidePullbackShape J) (j' : CategoryTheory.Limits.WidePullbackShape J) : Fintype (j ⟶ j')

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• One or more equations did not get rendered due to their size.

instance CategoryTheory.Limits.WidePushoutShape.fintypeObj {J : Type v} [Fintype J] : Fintype (CategoryTheory.Limits.WidePushoutShape J)

Equations

• CategoryTheory.Limits.WidePushoutShape.fintypeObj = Eq.mpr (_ : Fintype (CategoryTheory.Limits.WidePushoutShape J) = Fintype (Option J)) inferInstance

instance CategoryTheory.Limits.WidePushoutShape.fintypeHom {J : Type v} (j : CategoryTheory.Limits.WidePushoutShape J) (j' : CategoryTheory.Limits.WidePushoutShape J) : Fintype (j ⟶ j')

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• One or more equations did not get rendered due to their size.

instance CategoryTheory.Limits.finCategoryWidePullback {J : Type v} [Fintype J] : CategoryTheory.FinCategory (CategoryTheory.Limits.WidePullbackShape J)

Equations

• CategoryTheory.Limits.finCategoryWidePullback = CategoryTheory.FinCategory.mk

instance CategoryTheory.Limits.finCategoryWidePushout {J : Type v} [Fintype J] : CategoryTheory.FinCategory (CategoryTheory.Limits.WidePushoutShape J)

Equations

• CategoryTheory.Limits.finCategoryWidePushout = CategoryTheory.FinCategory.mk

class CategoryTheory.Limits.HasFiniteWidePullbacks (C : Type u) [CategoryTheory.Category.{v, u} C] : Prop

• out : ∀ (J : Type) [inst : Fintype J], CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.Limits.WidePullbackShape J) C

  C has all wide pullbacks any Fintype J

HasFiniteWidePullbacks represents a choice of wide pullback for every finite collection of morphisms

instance CategoryTheory.Limits.hasLimitsOfShape_widePullbackShape (C : Type u) [CategoryTheory.Category.{v, u} C] (J : Type) [Finite J] [CategoryTheory.Limits.HasFiniteWidePullbacks C] : CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.Limits.WidePullbackShape J) C

Equations

• CategoryTheory.Limits.hasLimitsOfShape_widePullbackShape = CategoryTheory.Limits.hasLimitsOfShape_widePullbackShape.proof_1

class CategoryTheory.Limits.HasFiniteWidePushouts (C : Type u) [CategoryTheory.Category.{v, u} C] : Prop

• out : ∀ (J : Type) [inst : Fintype J], CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.Limits.WidePushoutShape J) C

  C has all wide pushouts any Fintype J

HasFiniteWidePushouts represents a choice of wide pushout for every finite collection of morphisms

instance CategoryTheory.Limits.hasColimitsOfShape_widePushoutShape (C : Type u) [CategoryTheory.Category.{v, u} C] (J : Type) [Finite J] [CategoryTheory.Limits.HasFiniteWidePushouts C] : CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.Limits.WidePushoutShape J) C

Equations

• CategoryTheory.Limits.hasColimitsOfShape_widePushoutShape = CategoryTheory.Limits.hasColimitsOfShape_widePushoutShape.proof_1

theorem CategoryTheory.Limits.hasFiniteWidePullbacks_of_hasFiniteLimits (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteLimits C] : CategoryTheory.Limits.HasFiniteWidePullbacks C

Finite wide pullbacks are finite limits, so if C has all finite limits, it also has finite wide pullbacks

theorem CategoryTheory.Limits.hasFiniteWidePushouts_of_has_finite_limits (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteColimits C] : CategoryTheory.Limits.HasFiniteWidePushouts C

Finite wide pushouts are finite colimits, so if C has all finite colimits, it also has finite wide pushouts

instance CategoryTheory.Limits.fintypeWalkingPair : Fintype CategoryTheory.Limits.WalkingPair

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• One or more equations did not get rendered due to their size.