The category paths on a quiver.

When C is a quiver, paths C is the category of paths.

When the quiver is itself a category

We provide path_composition : paths C ⥤ C.

We check that the quotient of the path category of a category by the canonical relation (paths are related if they compose to the same path) is equivalent to the original category.

def CategoryTheory.Paths (V : Type u₁) : Type u₁

A type synonym for the category of paths in a quiver.

@[implicit_reducible]

instance CategoryTheory.instInhabitedPaths (V : Type u₁) [Inhabited V] : Inhabited (Paths V)

@[implicit_reducible]

instance CategoryTheory.instUniquePaths (V : Type u₁) [Unique V] : Unique (Paths V)

@[implicit_reducible]

instance CategoryTheory.Paths.categoryPaths (V : Type u₁) [Quiver V] : Category.{max u₁ v₁, u₁} (Paths V)

def CategoryTheory.Paths.of (V : Type u₁) [Quiver V] : V ⥤q Paths V

The inclusion of a quiver V into its path category, as a prefunctor.

@[simp]

theorem CategoryTheory.Paths.of_map (V : Type u₁) [Quiver V] {X✝ Y✝ : V} (f : X✝ ⟶ Y✝) : (of V).map f = f.toPath

@[simp]

theorem CategoryTheory.Paths.of_obj (V : Type u₁) [Quiver V] (X : V) : (of V).obj X = X

theorem CategoryTheory.Paths.induction_fixed_source {V : Type u₁} [Quiver V] {a : Paths V} (P : {b : Paths V} → (a ⟶ b) → Prop) (id : P (CategoryStruct.id a)) (comp : ∀ {u v : V} (p : a ⟶ (of V).obj u) (q : u ⟶ v), P p → P (CategoryStruct.comp p ((of V).map q))) {b : Paths V} (f : a ⟶ b) : P f

To prove a property on morphisms of a path category with given source a, it suffices to prove it for the identity and prove that the property is preserved under composition on the right with length 1 paths.

theorem CategoryTheory.Paths.induction_fixed_target {V : Type u₁} [Quiver V] {b : Paths V} (P : {a : Paths V} → (a ⟶ b) → Prop) (id : P (CategoryStruct.id b)) (comp : ∀ {u v : V} (p : (of V).obj v ⟶ b) (q : u ⟶ v), P p → P (CategoryStruct.comp ((of V).map q) p)) {a : Paths V} (f : a ⟶ b) : P f

To prove a property on morphisms of a path category with given target b, it suffices to prove it for the identity and prove that the property is preserved under composition on the left with length 1 paths.

theorem CategoryTheory.Paths.induction {V : Type u₁} [Quiver V] (P : {a b : Paths V} → (a ⟶ b) → Prop) (id : ∀ {v : V}, P (CategoryStruct.id ((of V).obj v))) (comp : ∀ {u v w : V} (p : (of V).obj u ⟶ (of V).obj v) (q : v ⟶ w), P p → P (CategoryStruct.comp p ((of V).map q))) {a b : Paths V} (f : a ⟶ b) : P f

To prove a property on morphisms of a path category, it suffices to prove it for the identity and prove that the property is preserved under composition on the right with length 1 paths.

theorem CategoryTheory.Paths.induction' {V : Type u₁} [Quiver V] (P : {a b : Paths V} → (a ⟶ b) → Prop) (id : ∀ {v : V}, P (CategoryStruct.id ((of V).obj v))) (comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P (CategoryStruct.comp ((of V).map p) q)) {a b : Paths V} (f : a ⟶ b) : P f

To prove a property on morphisms of a path category, it suffices to prove it for the identity and prove that the property is preserved under composition on the left with length 1 paths.

def CategoryTheory.Paths.lift {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] (φ : V ⥤q C) : Functor (Paths V) C

Any prefunctor from V lifts to a functor from paths V

@[simp]

theorem CategoryTheory.Paths.lift_nil {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] (φ : V ⥤q C) (X : V) : (lift φ).map Quiver.Path.nil = CategoryStruct.id (φ.obj X)

@[simp]

theorem CategoryTheory.Paths.lift_cons {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] (φ : V ⥤q C) {X Y Z : V} (p : Quiver.Path X Y) (f : Y ⟶ Z) : (lift φ).map (p.cons f) = CategoryStruct.comp ((lift φ).map p) (φ.map f)

@[simp]

theorem CategoryTheory.Paths.lift_toPath {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] (φ : V ⥤q C) {X Y : V} (f : X ⟶ Y) : (lift φ).map f.toPath = φ.map f

theorem CategoryTheory.Paths.lift_spec {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] (φ : V ⥤q C) : of V ⋙q (lift φ).toPrefunctor = φ

theorem CategoryTheory.Paths.lift_unique {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] (φ : V ⥤q C) (Φ : Functor (Paths V) C) (hΦ : of V ⋙q Φ.toPrefunctor = φ) : Φ = lift φ

theorem CategoryTheory.Paths.ext_functor {V : Type u₁} [Quiver V] {C : Type u_1} [Category.{v_1, u_1} C] {F G : Functor (Paths V) C} (h_obj : F.obj = G.obj) (h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = CategoryStruct.comp (eqToHom ⋯) (CategoryStruct.comp (G.map e.toPath) (eqToHom ⋯))) : F = G

Two functors out of a path category are equal when they agree on singleton paths.

@[simp]

theorem CategoryTheory.Prefunctor.mapPath_comp' (V : Type u₁) [Quiver V] (W : Type u₂) [Quiver W] (F : V ⥤q W) {X Y Z : Paths V} (f : X ⟶ Y) (g : Y ⟶ Z) : F.mapPath (CategoryStruct.comp f g) = (F.mapPath f).comp (F.mapPath g)

def CategoryTheory.composePath {C : Type u₁} [Category.{v₁, u₁} C] {X Y : C} : Quiver.Path X Y → (X ⟶ Y)

A path in a category can be composed to a single morphism.

theorem CategoryTheory.composePath_nil {C : Type u₁} [Category.{v₁, u₁} C] {X : C} : composePath Quiver.Path.nil = CategoryStruct.id X

theorem CategoryTheory.composePath_cons {C : Type u₁} [Category.{v₁, u₁} C] {X Y Z : C} (p : Quiver.Path X Y) (e : Y ⟶ Z) : composePath (p.cons e) = CategoryStruct.comp (composePath p) e

@[simp]

theorem CategoryTheory.composePath_toPath {C : Type u₁} [Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) : composePath f.toPath = f

@[simp]

theorem CategoryTheory.composePath_comp {C : Type u₁} [Category.{v₁, u₁} C] {X Y Z : C} (f : Quiver.Path X Y) (g : Quiver.Path Y Z) : composePath (f.comp g) = CategoryStruct.comp (composePath f) (composePath g)

@[simp]

theorem CategoryTheory.composePath_id {C : Type u₁} [Category.{v₁, u₁} C] {X : Paths C} : composePath (CategoryStruct.id X) = CategoryStruct.id (have this := X; this)

@[simp]

theorem CategoryTheory.composePath_comp' {C : Type u₁} [Category.{v₁, u₁} C] {X Y Z : Paths C} (f : X ⟶ Y) (g : Y ⟶ Z) : composePath (CategoryStruct.comp f g) = CategoryStruct.comp (composePath f) (composePath g)

def CategoryTheory.pathComposition (C : Type u₁) [Category.{v₁, u₁} C] : Functor (Paths C) C

Composition of paths as functor from the path category of a category to the category.

@[simp]

theorem CategoryTheory.pathComposition_obj (C : Type u₁) [Category.{v₁, u₁} C] (X : Paths C) : (pathComposition C).obj X = X

@[simp]

theorem CategoryTheory.pathComposition_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : Paths C} (f : X✝ ⟶ Y✝) : (pathComposition C).map f = composePath f

def CategoryTheory.pathsHomRel (C : Type u₁) [Category.{v₁, u₁} C] : HomRel (Paths C)

The canonical relation on the path category of a category: two paths are related if they compose to the same morphism.

def CategoryTheory.toQuotientPaths (C : Type u₁) [Category.{v₁, u₁} C] : Functor C (Quotient (pathsHomRel C))

The functor from a category to the canonical quotient of its path category.

@[simp]

theorem CategoryTheory.toQuotientPaths_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : C} (f : X✝ ⟶ Y✝) : (toQuotientPaths C).map f = Quot.mk (HomRel.CompClosure (pathsHomRel C)) f.toPath

@[simp]

theorem CategoryTheory.toQuotientPaths_obj_as (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : ((toQuotientPaths C).obj X).as = X

def CategoryTheory.quotientPathsTo (C : Type u₁) [Category.{v₁, u₁} C] : Functor (Quotient (pathsHomRel C)) C

The functor from the canonical quotient of a path category of a category to the original category.

@[simp]

theorem CategoryTheory.quotientPathsTo_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : Quotient (pathsHomRel C)} (hf : X✝ ⟶ Y✝) : (quotientPathsTo C).map hf = Quot.liftOn hf (fun (f : X✝.as ⟶ Y✝.as) => composePath f) ⋯

@[simp]

theorem CategoryTheory.quotientPathsTo_obj (C : Type u₁) [Category.{v₁, u₁} C] (a : Quotient (pathsHomRel C)) : (quotientPathsTo C).obj a = a.as

def CategoryTheory.quotientPathsEquiv (C : Type u₁) [Category.{v₁, u₁} C] : Quotient (pathsHomRel C) ≌ C

The canonical quotient of the path category of a category is equivalent to the original category.