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category_theory.is_connected

Connected category #

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Define a connected category as a nonempty category for which every functor to a discrete category is isomorphic to the constant functor.

NB. Some authors include the empty category as connected, we do not. We instead are interested in categories with exactly one 'connected component'.

We give some equivalent definitions:

A nonempty category for which every functor to a discrete category is constant on objects. See any_functor_const_on_obj and connected.of_any_functor_const_on_obj.
A nonempty category for which every function F for which the presence of a morphism f : j₁ ⟶ j₂ implies F j₁ = F j₂ must be constant everywhere. See constant_of_preserves_morphisms and connected.of_constant_of_preserves_morphisms.
A nonempty category for which any subset of its elements containing the default and closed under morphisms is everything. See induct_on_objects and connected.of_induct.
A nonempty category for which every object is related under the reflexive transitive closure of the relation "there is a morphism in some direction from j₁ to j₂". See connected_zigzag and zigzag_connected.
A nonempty category for which for any two objects there is a sequence of morphisms (some reversed) from one to the other. See exists_zigzag' and connected_of_zigzag.

We also prove the result that the functor given by (X × -) preserves any connected limit. That is, any limit of shape J where J is a connected category is preserved by the functor (X × -). This appears in category_theory.limits.connected.

@[class]

structure category_theory.is_preconnected (J : Type u₁) [category_theory.category J] :

Prop

iso_constant : ∀ {α : Type ?} (F : J ⥤ category_theory.discrete α) (j : J), nonempty (F ≅ (category_theory.functor.const J).obj (F.obj j))

A possibly empty category for which every functor to a discrete category is constant.

Instances of this typeclass

@[class]

structure category_theory.is_connected (J : Type u₁) [category_theory.category J] :

Prop

to_is_preconnected : category_theory.is_preconnected J
is_nonempty : nonempty J

We define a connected category as a nonempty category for which every functor to a discrete category is constant.

NB. Some authors include the empty category as connected, we do not. We instead are interested in categories with exactly one 'connected component'.

This allows us to show that the functor X ⨯ - preserves connected limits.

See https://stacks.math.columbia.edu/tag/002S

Instances of this typeclass

noncomputable def category_theory.iso_constant {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected J] {α : Type u₁} (F : J ⥤ category_theory.discrete α) (j : J) :

F ≅ (category_theory.functor.const J).obj (F.obj j)

If J is connected, any functor F : J ⥤ discrete α is isomorphic to the constant functor with value F.obj j (for any choice of j).

Equations

category_theory.iso_constant F j = _.some

theorem category_theory.any_functor_const_on_obj {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected J] {α : Type u₁} (F : J ⥤ category_theory.discrete α) (j j' : J) :

F.obj j = F.obj j'

If J is connected, any functor to a discrete category is constant on objects. The converse is given in is_connected.of_any_functor_const_on_obj.

theorem category_theory.is_connected.of_any_functor_const_on_obj {J : Type u₁} [category_theory.category J] [nonempty J] (h : ∀ {α : Type u₁} (F : J ⥤ category_theory.discrete α) (j j' : J), F.obj j = F.obj j') :

category_theory.is_connected J

If any functor to a discrete category is constant on objects, J is connected. The converse of any_functor_const_on_obj.

theorem category_theory.constant_of_preserves_morphisms {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected J] {α : Type u₁} (F : J → α) (h : ∀ (j₁ j₂ : J), (j₁ ⟶ j₂) → F j₁ = F j₂) (j j' : J) :

F j = F j'

If J is connected, then given any function F such that the presence of a morphism j₁ ⟶ j₂ implies F j₁ = F j₂, we have that F is constant. This can be thought of as a local-to-global property.

The converse is shown in is_connected.of_constant_of_preserves_morphisms

theorem category_theory.is_connected.of_constant_of_preserves_morphisms {J : Type u₁} [category_theory.category J] [nonempty J] (h : ∀ {α : Type u₁} (F : J → α), (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂) → ∀ (j j' : J), F j = F j') :

category_theory.is_connected J

J is connected if: given any function F : J → α which is constant for any j₁, j₂ for which there is a morphism j₁ ⟶ j₂, then F is constant. This can be thought of as a local-to-global property.

The converse of constant_of_preserves_morphisms.

theorem category_theory.induct_on_objects {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected J] (p : set J) {j₀ : J} (h0 : j₀ ∈ p) (h1 : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) (j : J) :

j ∈ p

An inductive-like property for the objects of a connected category. If the set p is nonempty, and p is closed under morphisms of J, then p contains all of J.

The converse is given in is_connected.of_induct.

theorem category_theory.is_connected.of_induct {J : Type u₁} [category_theory.category J] [nonempty J] {j₀ : J} (h : ∀ (p : set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p) :

category_theory.is_connected J

If any maximal connected component containing some element j₀ of J is all of J, then J is connected.

The converse of induct_on_objects.

@[protected, instance]

def category_theory.ulift_hom.is_connected {J : Type u₁} [category_theory.category J] [hc : category_theory.is_connected J] :

category_theory.is_connected (category_theory.ulift_hom (ulift J))

Lifting the universe level of morphisms and objects preserves connectedness.

theorem category_theory.is_preconnected_induction {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected J] (Z : J → Sort u_1) (h₁ : Π {j₁ j₂ : J}, (j₁ ⟶ j₂) → Z j₁ → Z j₂) (h₂ : Π {j₁ j₂ : J}, (j₁ ⟶ j₂) → Z j₂ → Z j₁) {j₀ : J} (x : Z j₀) (j : J) :

Another induction principle for is_preconnected J: given a type family Z : J → Sort* and a rule for transporting in both directions along a morphism in J, we can transport an x : Z j₀ to a point in Z j for any j.

theorem category_theory.is_preconnected_of_equivalent {J : Type u₁} [category_theory.category J] {K : Type u₁} [category_theory.category K] [category_theory.is_preconnected J] (e : J ≌ K) :

category_theory.is_preconnected K

If J and K are equivalent, then if J is preconnected then K is as well.

theorem category_theory.is_connected_of_equivalent {J : Type u₁} [category_theory.category J] {K : Type u₁} [category_theory.category K] (e : J ≌ K) [category_theory.is_connected J] :

category_theory.is_connected K

If J and K are equivalent, then if J is connected then K is as well.

@[protected, instance]

def category_theory.is_preconnected_op {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected J] :

category_theory.is_preconnected Jᵒᵖ

If J is preconnected, then Jᵒᵖ is preconnected as well.

@[protected, instance]

def category_theory.is_connected_op {J : Type u₁} [category_theory.category J] [category_theory.is_connected J] :

category_theory.is_connected Jᵒᵖ

If J is connected, then Jᵒᵖ is connected as well.

theorem category_theory.is_preconnected_of_is_preconnected_op {J : Type u₁} [category_theory.category J] [category_theory.is_preconnected Jᵒᵖ] :

category_theory.is_preconnected J

theorem category_theory.is_connected_of_is_connected_op {J : Type u₁} [category_theory.category J] [category_theory.is_connected Jᵒᵖ] :

category_theory.is_connected J

@[reducible]

def category_theory.zag {J : Type u₁} [category_theory.category J] (j₁ j₂ : J) :

Prop

j₁ and j₂ are related by zag if there is a morphism between them.

Equations

category_theory.zag j₁ j₂ = (nonempty (j₁ ⟶ j₂) ∨ nonempty (j₂ ⟶ j₁))

theorem category_theory.zag_symmetric {J : Type u₁} [category_theory.category J] :

symmetric category_theory.zag

@[reducible]

def category_theory.zigzag {J : Type u₁} [category_theory.category J] :

J → J → Prop

j₁ and j₂ are related by zigzag if there is a chain of morphisms from j₁ to j₂, with backward morphisms allowed.

Equations

category_theory.zigzag = relation.refl_trans_gen category_theory.zag

theorem category_theory.zigzag_symmetric {J : Type u₁} [category_theory.category J] :

symmetric category_theory.zigzag

theorem category_theory.zigzag_equivalence {J : Type u₁} [category_theory.category J] :

equivalence category_theory.zigzag

def category_theory.zigzag.setoid (J : Type u₂) [category_theory.category J] :

The setoid given by the equivalence relation zigzag. A quotient for this setoid is a connected component of the category.

Equations

category_theory.zigzag.setoid J = {r := category_theory.zigzag _inst_3, iseqv := _}

theorem category_theory.zigzag_obj_of_zigzag {J : Type u₁} [category_theory.category J] {K : Type u₂} [category_theory.category K] (F : J ⥤ K) {j₁ j₂ : J} (h : category_theory.zigzag j₁ j₂) :

category_theory.zigzag (F.obj j₁) (F.obj j₂)

If there is a zigzag from j₁ to j₂, then there is a zigzag from F j₁ to F j₂ as long as F is a functor.

theorem category_theory.zag_of_zag_obj {J : Type u₁} [category_theory.category J] {K : Type u₂} [category_theory.category K] (F : J ⥤ K) [category_theory.full F] {j₁ j₂ : J} (h : category_theory.zag (F.obj j₁) (F.obj j₂)) :

category_theory.zag j₁ j₂

theorem category_theory.equiv_relation {J : Type u₁} [category_theory.category J] [category_theory.is_connected J] (r : J → J → Prop) (hr : equivalence r) (h : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → r j₁ j₂) (j₁ j₂ : J) :

r j₁ j₂

Any equivalence relation containing (⟶) holds for all pairs of a connected category.

theorem category_theory.is_connected_zigzag {J : Type u₁} [category_theory.category J] [category_theory.is_connected J] (j₁ j₂ : J) :

category_theory.zigzag j₁ j₂

In a connected category, any two objects are related by zigzag.

theorem category_theory.zigzag_is_connected {J : Type u₁} [category_theory.category J] [nonempty J] (h : ∀ (j₁ j₂ : J), category_theory.zigzag j₁ j₂) :

category_theory.is_connected J

If any two objects in an nonempty category are related by zigzag, the category is connected.

theorem category_theory.exists_zigzag' {J : Type u₁} [category_theory.category J] [category_theory.is_connected J] (j₁ j₂ : J) :

∃ (l : list J), list.chain category_theory.zag j₁ l ∧ (j₁ :: l).last _ = j₂

theorem category_theory.is_connected_of_zigzag {J : Type u₁} [category_theory.category J] [nonempty J] (h : ∀ (j₁ j₂ : J), ∃ (l : list J), list.chain category_theory.zag j₁ l ∧ (j₁ :: l).last _ = j₂) :

category_theory.is_connected J

If any two objects in an nonempty category are linked by a sequence of (potentially reversed) morphisms, then J is connected.

The converse of exists_zigzag'.

noncomputable def category_theory.discrete_is_connected_equiv_punit {α : Type u₁} [category_theory.is_connected (category_theory.discrete α)] :

If discrete α is connected, then α is (type-)equivalent to punit.

Equations

theorem category_theory.nat_trans_from_is_connected {J : Type u₁} [category_theory.category J] {C : Type u₂} [category_theory.category C] [category_theory.is_preconnected J] {X Y : C} (α : (category_theory.functor.const J).obj X ⟶ (category_theory.functor.const J).obj Y) (j j' : J) :

α.app j = α.app j'

For objects X Y : C, any natural transformation α : const X ⟶ const Y from a connected category must be constant. This is the key property of connected categories which we use to establish properties about limits.

@[protected, instance]

noncomputable def category_theory.functor.const.full {J : Type u₁} [category_theory.category J] {C : Type u₂} [category_theory.category C] [category_theory.is_connected J] :

category_theory.full (category_theory.functor.const J)

Equations

category_theory.functor.const.full = {preimage := λ (X Y : C) (f : (category_theory.functor.const J).obj X ⟶ (category_theory.functor.const J).obj Y), f.app (classical.arbitrary J), witness' := _}

@[protected, instance]

def category_theory.nonempty_hom_of_connected_groupoid {G : Type u_1} [category_theory.groupoid G] [category_theory.is_connected G] (x y : G) :

nonempty (x ⟶ y)