mathlib documentation

category_theory.filtered

Filtered categories #

A category is filtered if every finite diagram admits a cocone. We give a simple characterisation of this condition as

  1. for every pair of objects there exists another object "to the right",
  2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal, and
  3. there exists some object.

Filtered colimits are often better behaved than arbitrary colimits. See category_theory/limits/types for some details.

Filtered categories are nice because colimits indexed by filtered categories tend to be easier to describe than general colimits (and more often preserved by functors).

In this file we show that any functor from a finite category to a filtered category admits a cocone:

We also provide all of the above API for cofiltered categories.

See also #

In category_theory.limits.filtered_colimit_commutes_finite_limit we show that filtered colimits commute with finite limits.

Future work #

@[class]
  • cocone_objs : ∀ (X Y : C), ∃ (Z : C) (f : X Z) (g : Y Z), true
  • cocone_maps : ∀ ⦃X Y : C⦄ (f g : X Y), ∃ (Z : C) (h : Y Z), f h = g h

A category is_filtered_or_empty if

  1. for every pair of objects there exists another object "to the right", and
  2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal.
Instances
@[class]
structure category_theory.is_filtered (C : Type u) [category_theory.category C] :
Prop

A category is_filtered if

  1. for every pair of objects there exists another object "to the right",
  2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal, and
  3. there exists some object.

See https://stacks.math.columbia.edu/tag/002V. (They also define a diagram being filtered.)

Instances

max j j' is an arbitrary choice of object to the right of both j and j', whose existence is ensured by is_filtered.

Equations

left_to_max j j' is an arbitrarily choice of morphism from j to max j j', whose existence is ensured by is_filtered.

Equations

right_to_max j j' is an arbitrarily choice of morphism from j' to max j j', whose existence is ensured by is_filtered.

Equations
def category_theory.is_filtered.coeq {C : Type u} [category_theory.category C] [category_theory.is_filtered C] {j j' : C} (f f' : j j') :
C

coeq f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of object which admits a morphism coeq_hom f f' : j' ⟶ coeq f f' such that coeq_condition : f ≫ coeq_hom f f' = f' ≫ coeq_hom f f'. Its existence is ensured by is_filtered.

Equations

coeq_hom f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of morphism coeq_hom f f' : j' ⟶ coeq f f' such that coeq_condition : f ≫ coeq_hom f f' = f' ≫ coeq_hom f f'. Its existence is ensured by is_filtered.

Equations
@[simp]

coeq_condition f f', for morphisms f f' : j ⟶ j', is the proof that f ≫ coeq_hom f f' = f' ≫ coeq_hom f f'.

theorem category_theory.is_filtered.sup_objs_exists {C : Type u} [category_theory.category C] [category_theory.is_filtered C] (O : finset C) :
∃ (S : C), ∀ {X : C}, X Ononempty (X S)

Any finite collection of objects in a filtered category has an object "to the right".

theorem category_theory.is_filtered.sup_exists {C : Type u} [category_theory.category C] [category_theory.is_filtered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) :
∃ (S : C) (T : Π {X : C}, X O(X S)), ∀ {X Y : C} (mX : X O) (mY : Y O) {f : X Y}, X, Y, mX, mY, f⟩ Hf T mY = T mX

Given any finset of objects {X, ...} and indexed collection of finsets of morphisms {f, ...} in C, there exists an object S, with a morphism T X : X ⟶ S from each X, such that the triangles commute: f ≫ T Y = T X, for f : X ⟶ Y in the finset.

def category_theory.is_filtered.sup {C : Type u} [category_theory.category C] [category_theory.is_filtered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) :
C

An arbitrary choice of object "to the right" of a finite collection of objects O and morphisms H, making all the triangles commute.

Equations
def category_theory.is_filtered.to_sup {C : Type u} [category_theory.category C] [category_theory.is_filtered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) {X : C} (m : X O) :

The morphisms to sup O H.

Equations
theorem category_theory.is_filtered.to_sup_commutes {C : Type u} [category_theory.category C] [category_theory.is_filtered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) {X Y : C} (mX : X O) (mY : Y O) {f : X Y} (mf : X, Y, mX, mY, f⟩ H) :

The triangles of consisting of a morphism in H and the maps to sup O H commute.

If we have is_filtered C, then for any functor F : J ⥤ C with fin_category J, there exists a cocone over F.

An arbitrary choice of cocone over F : J ⥤ C, for fin_category J and is_filtered C.

Equations

If C is filtered, and we have a functor R : C ⥤ D with a left adjoint, then D is filtered.

If C is filtered, and we have a right adjoint functor R : C ⥤ D, then D is filtered.

Being filtered is preserved by equivalence of categories.

@[class]
  • cocone_objs : ∀ (X Y : C), ∃ (W : C) (f : W X) (g : W Y), true
  • cocone_maps : ∀ ⦃X Y : C⦄ (f g : X Y), ∃ (W : C) (h : W X), h f = h g

A category is_cofiltered_or_empty if

  1. for every pair of objects there exists another object "to the left", and
  2. for every pair of parallel morphisms there exists a morphism to the left so the compositions are equal.
Instances
@[class]
structure category_theory.is_cofiltered (C : Type u) [category_theory.category C] :
Prop

A category is_cofiltered if

  1. for every pair of objects there exists another object "to the left",
  2. for every pair of parallel morphisms there exists a morphism to the left so the compositions are equal, and
  3. there exists some object.

See https://stacks.math.columbia.edu/tag/04AZ.

Instances

min j j' is an arbitrary choice of object to the left of both j and j', whose existence is ensured by is_cofiltered.

Equations

min_to_left j j' is an arbitrarily choice of morphism from min j j' to j, whose existence is ensured by is_cofiltered.

Equations

min_to_right j j' is an arbitrarily choice of morphism from min j j' to j', whose existence is ensured by is_cofiltered.

Equations
def category_theory.is_cofiltered.eq {C : Type u} [category_theory.category C] [category_theory.is_cofiltered C] {j j' : C} (f f' : j j') :
C

eq f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of object which admits a morphism eq_hom f f' : eq f f' ⟶ j such that eq_condition : eq_hom f f' ≫ f = eq_hom f f' ≫ f'. Its existence is ensured by is_cofiltered.

Equations

eq_hom f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of morphism eq_hom f f' : eq f f' ⟶ j such that eq_condition : eq_hom f f' ≫ f = eq_hom f f' ≫ f'. Its existence is ensured by is_cofiltered.

Equations
@[simp]

eq_condition f f', for morphisms f f' : j ⟶ j', is the proof that eq_hom f f' ≫ f = eq_hom f f' ≫ f'.

theorem category_theory.is_cofiltered.inf_objs_exists {C : Type u} [category_theory.category C] [category_theory.is_cofiltered C] (O : finset C) :
∃ (S : C), ∀ {X : C}, X Ononempty (S X)

Any finite collection of objects in a cofiltered category has an object "to the left".

theorem category_theory.is_cofiltered.inf_exists {C : Type u} [category_theory.category C] [category_theory.is_cofiltered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) :
∃ (S : C) (T : Π {X : C}, X O(S X)), ∀ {X Y : C} (mX : X O) (mY : Y O) {f : X Y}, X, Y, mX, mY, f⟩ HT mX f = T mY

Given any finset of objects {X, ...} and indexed collection of finsets of morphisms {f, ...} in C, there exists an object S, with a morphism T X : S ⟶ X from each X, such that the triangles commute: T X ≫ f = T Y, for f : X ⟶ Y in the finset.

def category_theory.is_cofiltered.inf {C : Type u} [category_theory.category C] [category_theory.is_cofiltered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) :
C

An arbitrary choice of object "to the left" of a finite collection of objects O and morphisms H, making all the triangles commute.

Equations
def category_theory.is_cofiltered.inf_to {C : Type u} [category_theory.category C] [category_theory.is_cofiltered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) {X : C} (m : X O) :

The morphisms from inf O H.

Equations
theorem category_theory.is_cofiltered.inf_to_commutes {C : Type u} [category_theory.category C] [category_theory.is_cofiltered C] (O : finset C) (H : finset (Σ' (X Y : C) (mX : X O) (mY : Y O), X Y)) {X Y : C} (mX : X O) (mY : Y O) {f : X Y} (mf : X, Y, mX, mY, f⟩ H) :

The triangles consisting of a morphism in H and the maps from inf O H commute.

If we have is_cofiltered C, then for any functor F : J ⥤ C with fin_category J, there exists a cone over F.

An arbitrary choice of cone over F : J ⥤ C, for fin_category J and is_cofiltered C.

Equations

If C is cofiltered, and we have a functor L : C ⥤ D with a right adjoint, then D is cofiltered.

If C is cofiltered, and we have a left adjoint functor L : C ⥤ D, then D is cofiltered.

Being cofiltered is preserved by equivalence of categories.