Categories

Defines a category, as a type class parametrised by the type of objects.

Notations

Introduces notations in the CategoryTheory scope

• X ⟶ Y for the morphism spaces (type as \hom),
• 𝟙 X for the identity morphism on X (type as \b1),
• f ≫ g for composition in the 'arrows' convention (type as \gg).

Users may like to add g ⊚ f for composition in the standard convention, using

local notation g ` ⊚ `:80 f:80 := category.comp f g    -- type as \oo

Porting note

I am experimenting with using the aesop tactic as a replacement for tidy.

class CategoryTheory.CategoryStruct (obj : Type u) extends Quiver : Type (max u (v + 1))

A preliminary structure on the way to defining a category, containing the data, but none of the axioms.

• Hom : obj → obj → Type v
• id : (X : obj) → X ⟶ X

  The identity morphism on an object.

• comp : {X Y Z : obj} → (X ⟶ Y) → (Y ⟶ Z) → (X ⟶ Z)

  Composition of morphisms in a category, written f ≫ g.

def CategoryTheory.«term𝟙» : Lean.ParserDescr

Notation for the identity morphism in a category.

Equations

• CategoryTheory.«term𝟙» = Lean.ParserDescr.node `CategoryTheory.term𝟙 1024 (Lean.ParserDescr.symbol "𝟙")

def CategoryTheory.«term_≫_» : Lean.TrailingParserDescr

Notation for composition of morphisms in a category.

Equations

• CategoryTheory.«term_≫_» = Lean.ParserDescr.trailingNode `CategoryTheory.term_≫_ 80 81 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≫ ") (Lean.ParserDescr.cat `term 80))

def CategoryTheory.sorryIfSorry : Lean.ParserDescr

Close the main goal with sorry if its type contains sorry, and fail otherwise.

Equations

• CategoryTheory.sorryIfSorry = Lean.ParserDescr.node `CategoryTheory.sorryIfSorry 1024 (Lean.ParserDescr.nonReservedSymbol "sorry_if_sorry" false)

def CategoryTheory.evalSorryIfSorry : Lean.Elab.Tactic.Tactic

Close the main goal with sorry if its type contains sorry, and fail otherwise.

Equations

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

def CategoryTheory.aesop_cat : Lean.ParserDescr

A thin wrapper for aesop which adds the CategoryTheory rule set and allows aesop to look through semireducible definitions when calling intros. This tactic fails when it is unable to solve the goal, making it suitable for use in auto-params.

Equations

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

def CategoryTheory.aesop_cat? : Lean.ParserDescr

We also use aesop_cat? to pass along a Try this suggestion when using aesop_cat

Equations

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

def CategoryTheory.aesop_cat_nonterminal : Lean.ParserDescr

A variant of aesop_cat which does not fail when it is unable to solve the goal. Use this only for exploration! Nonterminal aesop is even worse than nonterminal simp.

Equations

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

class CategoryTheory.Category (obj : Type u) extends CategoryTheory.CategoryStruct : Type (max u (v + 1))

The typeclass Category C describes morphisms associated to objects of type C. The universe levels of the objects and morphisms are unconstrained, and will often need to be specified explicitly, as Category.{v} C. (See also LargeCategory and SmallCategory.)

See https://stacks.math.columbia.edu/tag/0014.

• Hom : obj → obj → Type v
• id : (X : obj) → X ⟶ X
• comp : {X Y Z : obj} → (X ⟶ Y) → (Y ⟶ Z) → (X ⟶ Z)
• id_comp : ∀ {X Y : obj} (f : X ⟶ Y), CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id X) f = f

  Identity morphisms are left identities for composition.

• comp_id : ∀ {X Y : obj} (f : X ⟶ Y), CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.id Y) = f

  Identity morphisms are right identities for composition.

• assoc : ∀ {W X Y Z : obj} (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z), CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h)

  Composition in a category is associative.

@[simp]

theorem CategoryTheory.Category.id_comp {obj : Type u} [self : CategoryTheory.Category.{v, u} obj] {X : obj} {Y : obj} (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id X) f = f

Identity morphisms are left identities for composition.

@[simp]

theorem CategoryTheory.Category.comp_id {obj : Type u} [self : CategoryTheory.Category.{v, u} obj] {X : obj} {Y : obj} (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.id Y) = f

Identity morphisms are right identities for composition.

@[simp]

theorem CategoryTheory.Category.assoc {obj : Type u} [self : CategoryTheory.Category.{v, u} obj] {W : obj} {X : obj} {Y : obj} {Z : obj} (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h)

Composition in a category is associative.

@[reducible, inline]

abbrev CategoryTheory.LargeCategory (C : Type (u + 1)) : Type (u + 1)

A LargeCategory has objects in one universe level higher than the universe level of the morphisms. It is useful for examples such as the category of types, or the category of groups, etc.

Equations

• CategoryTheory.LargeCategory C = CategoryTheory.Category.{?u.10, ?u.10 + 1} C

@[reducible, inline]

abbrev CategoryTheory.SmallCategory (C : Type u) : Type (u + 1)

A SmallCategory has objects and morphisms in the same universe level.

Equations

• CategoryTheory.SmallCategory C = CategoryTheory.Category.{?u.10, ?u.10} C

theorem CategoryTheory.eq_whisker {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} {f : X ⟶ Y} {g : X ⟶ Y} (w : f = g) (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g h

postcompose an equation between morphisms by another morphism

theorem CategoryTheory.whisker_eq {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} {h : Y ⟶ Z} (w : g = h) : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f h

precompose an equation between morphisms by another morphism

def CategoryTheory.«term_=≫_» : Lean.TrailingParserDescr

Notation for whiskering an equation by a morphism (on the right). If f g : X ⟶ Y and w : f = g and h : Y ⟶ Z, then w =≫ h : f ≫ h = g ≫ h.

Equations

• CategoryTheory.«term_=≫_» = Lean.ParserDescr.trailingNode `CategoryTheory.term_=≫_ 80 81 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " =≫ ") (Lean.ParserDescr.cat `term 80))

def CategoryTheory.«term_≫=_» : Lean.TrailingParserDescr

Notation for whiskering an equation by a morphism (on the left). If g h : Y ⟶ Z and w : g = h and h : X ⟶ Y, then f ≫= w : f ≫ g = f ≫ h.

Equations

• CategoryTheory.«term_≫=_» = Lean.ParserDescr.trailingNode `CategoryTheory.term_≫=_ 80 81 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≫= ") (Lean.ParserDescr.cat `term 80))

theorem CategoryTheory.eq_of_comp_left_eq {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X ⟶ Y} {g : X ⟶ Y} (w : ∀ {Z : C} (h : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g h) : f = g

theorem CategoryTheory.eq_of_comp_right_eq {C : Type u} [CategoryTheory.Category.{v, u} C] {Y : C} {Z : C} {f : Y ⟶ Z} {g : Y ⟶ Z} (w : ∀ {X : C} (h : X ⟶ Y), CategoryTheory.CategoryStruct.comp h f = CategoryTheory.CategoryStruct.comp h g) : f = g

theorem CategoryTheory.eq_of_comp_left_eq' {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X ⟶ Y) (g : X ⟶ Y) (w : (fun {Z : C} (h : Y ⟶ Z) => CategoryTheory.CategoryStruct.comp f h) = fun {Z : C} (h : Y ⟶ Z) => CategoryTheory.CategoryStruct.comp g h) : f = g

theorem CategoryTheory.eq_of_comp_right_eq' {C : Type u} [CategoryTheory.Category.{v, u} C] {Y : C} {Z : C} (f : Y ⟶ Z) (g : Y ⟶ Z) (w : (fun {X : C} (h : X ⟶ Y) => CategoryTheory.CategoryStruct.comp h f) = fun {X : C} (h : X ⟶ Y) => CategoryTheory.CategoryStruct.comp h g) : f = g

theorem CategoryTheory.id_of_comp_left_id {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} (f : X ⟶ X) (w : ∀ {Y : C} (g : X ⟶ Y), CategoryTheory.CategoryStruct.comp f g = g) : f = CategoryTheory.CategoryStruct.id X

theorem CategoryTheory.id_of_comp_right_id {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} (f : X ⟶ X) (w : ∀ {Y : C} (g : Y ⟶ X), CategoryTheory.CategoryStruct.comp g f = g) : f = CategoryTheory.CategoryStruct.id X

theorem CategoryTheory.comp_ite {C : Type u} [CategoryTheory.Category.{v, u} C] {P : Prop} [Decidable P] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (g' : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp f (if P then g else g') = if P then CategoryTheory.CategoryStruct.comp f g else CategoryTheory.CategoryStruct.comp f g'

theorem CategoryTheory.ite_comp {C : Type u} [CategoryTheory.Category.{v, u} C] {P : Prop} [Decidable P] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (f' : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (if P then f else f') g = if P then CategoryTheory.CategoryStruct.comp f g else CategoryTheory.CategoryStruct.comp f' g

theorem CategoryTheory.comp_dite {C : Type u} [CategoryTheory.Category.{v, u} C] {P : Prop} [Decidable P] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : P → (Y ⟶ Z)) (g' : ¬P → (Y ⟶ Z)) : CategoryTheory.CategoryStruct.comp f (if h : P then g h else g' h) = if h : P then CategoryTheory.CategoryStruct.comp f (g h) else CategoryTheory.CategoryStruct.comp f (g' h)

theorem CategoryTheory.dite_comp {C : Type u} [CategoryTheory.Category.{v, u} C] {P : Prop} [Decidable P] {X : C} {Y : C} {Z : C} (f : P → (X ⟶ Y)) (f' : ¬P → (X ⟶ Y)) (g : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (if h : P then f h else f' h) g = if h : P then CategoryTheory.CategoryStruct.comp (f h) g else CategoryTheory.CategoryStruct.comp (f' h) g

class CategoryTheory.Epi {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X ⟶ Y) : Prop

A morphism f is an epimorphism if it can be cancelled when precomposed: f ≫ g = f ≫ h implies g = h.

See https://stacks.math.columbia.edu/tag/003B.

• left_cancellation : ∀ {Z : C} (g h : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f h → g = h

  A morphism f is an epimorphism if it can be cancelled when precomposed.

theorem CategoryTheory.Epi.left_cancellation {C : Type u} : ∀ {inst : CategoryTheory.Category.{v, u} C} {X Y : C} {f : X ⟶ Y} [self : CategoryTheory.Epi f] {Z : C} (g h : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f h → g = h

A morphism f is an epimorphism if it can be cancelled when precomposed.

class CategoryTheory.Mono {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X ⟶ Y) : Prop

A morphism f is a monomorphism if it can be cancelled when postcomposed: g ≫ f = h ≫ f implies g = h.

See https://stacks.math.columbia.edu/tag/003B.

• right_cancellation : ∀ {Z : C} (g h : Z ⟶ X), CategoryTheory.CategoryStruct.comp g f = CategoryTheory.CategoryStruct.comp h f → g = h

  A morphism f is a monomorphism if it can be cancelled when postcomposed.

theorem CategoryTheory.Mono.right_cancellation {C : Type u} : ∀ {inst : CategoryTheory.Category.{v, u} C} {X Y : C} {f : X ⟶ Y} [self : CategoryTheory.Mono f] {Z : C} (g h : Z ⟶ X), CategoryTheory.CategoryStruct.comp g f = CategoryTheory.CategoryStruct.comp h f → g = h

A morphism f is a monomorphism if it can be cancelled when postcomposed.

instance CategoryTheory.instEpiId {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) : CategoryTheory.Epi (CategoryTheory.CategoryStruct.id X)

Equations

• ⋯ = ⋯

instance CategoryTheory.instMonoId {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) : CategoryTheory.Mono (CategoryTheory.CategoryStruct.id X)

Equations

• ⋯ = ⋯

theorem CategoryTheory.cancel_epi {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) [CategoryTheory.Epi f] {g : Y ⟶ Z} {h : Y ⟶ Z} : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f h ↔ g = h

theorem CategoryTheory.cancel_epi_assoc_iff {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) [CategoryTheory.Epi f] {g : Y ⟶ Z} {h : Y ⟶ Z} {W : C} {k : Z ⟶ W} {l : Z ⟶ W} : CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) k = CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f h) l ↔ CategoryTheory.CategoryStruct.comp g k = CategoryTheory.CategoryStruct.comp h l

theorem CategoryTheory.cancel_mono {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) [CategoryTheory.Mono f] {g : Z ⟶ X} {h : Z ⟶ X} : CategoryTheory.CategoryStruct.comp g f = CategoryTheory.CategoryStruct.comp h f ↔ g = h

theorem CategoryTheory.cancel_mono_assoc_iff {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) [CategoryTheory.Mono f] {g : Z ⟶ X} {h : Z ⟶ X} {W : C} {k : W ⟶ Z} {l : W ⟶ Z} : CategoryTheory.CategoryStruct.comp k (CategoryTheory.CategoryStruct.comp g f) = CategoryTheory.CategoryStruct.comp l (CategoryTheory.CategoryStruct.comp h f) ↔ CategoryTheory.CategoryStruct.comp k g = CategoryTheory.CategoryStruct.comp l h

theorem CategoryTheory.cancel_epi_id {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] {h : Y ⟶ Y} : CategoryTheory.CategoryStruct.comp f h = f ↔ h = CategoryTheory.CategoryStruct.id Y

theorem CategoryTheory.cancel_mono_id {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] {g : X ⟶ X} : CategoryTheory.CategoryStruct.comp g f = f ↔ g = CategoryTheory.CategoryStruct.id X

instance CategoryTheory.epi_comp {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) [CategoryTheory.Epi f] (g : Y ⟶ Z) [CategoryTheory.Epi g] : CategoryTheory.Epi (CategoryTheory.CategoryStruct.comp f g)

Equations

• ⋯ = ⋯

instance CategoryTheory.mono_comp {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) [CategoryTheory.Mono f] (g : Y ⟶ Z) [CategoryTheory.Mono g] : CategoryTheory.Mono (CategoryTheory.CategoryStruct.comp f g)

Equations

• ⋯ = ⋯

theorem CategoryTheory.mono_of_mono {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.Mono (CategoryTheory.CategoryStruct.comp f g)] : CategoryTheory.Mono f

theorem CategoryTheory.mono_of_mono_fac {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [CategoryTheory.Mono h] (w : CategoryTheory.CategoryStruct.comp f g = h) : CategoryTheory.Mono f

theorem CategoryTheory.epi_of_epi {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.Epi (CategoryTheory.CategoryStruct.comp f g)] : CategoryTheory.Epi g

theorem CategoryTheory.epi_of_epi_fac {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [CategoryTheory.Epi h] (w : CategoryTheory.CategoryStruct.comp f g = h) : CategoryTheory.Epi g

instance CategoryTheory.instMono {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} [Quiver.IsThin C] (f : X ⟶ Y) : CategoryTheory.Mono f

Equations

• ⋯ = ⋯

instance CategoryTheory.instEpi {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} [Quiver.IsThin C] (f : X ⟶ Y) : CategoryTheory.Epi f

Equations

• ⋯ = ⋯

instance CategoryTheory.uliftCategory (C : Type u) [CategoryTheory.Category.{v, u} C] : CategoryTheory.Category.{v, max u' u} (ULift.{u', u} C)

Equations

• CategoryTheory.uliftCategory C = CategoryTheory.Category.mk ⋯ ⋯ ⋯