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:80 g " ⊚ " f:80 => CategoryTheory.CategoryStruct.comp f g    -- type as \oo

class CategoryTheory.CategoryStruct (obj : Type u) extends Quiver obj : 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

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def CategoryTheory.rfl_cat : Lean.ParserDescr

rfl_cat is a macro for intros; rfl which is attempted in aesop_cat before doing the more expensive aesop tactic.

This gives a speedup because simp (called by aesop) is too slow. There is a fix for this slowness in https://github.com/leanprover/lean4/pull/7428. So, when that is resolved, the performance impact of rfl_cat should be measured again.

Implementation notes:

• refine id ?_: In some cases it is important that the type of the proof matches the expected type exactly. e.g. if the goal is 2 = 1 + 1, the rfl tactic will give a proof of type 2 = 2. Starting a proof with refine id ?_ is a trick to make sure that the proof has exactly the expected type, in this case 2 = 1 + 1. See also https://leanprover.zulipchat.com/#narrow/channel/270676-lean4/topic/changing.20a.20proof.20can.20break.20a.20later.20proof
• apply_rfl: rfl is a macro that attempts both eq_refl and apply_rfl. Since apply_rfl subsumes eq_refl, we can use apply_rfl instead. This fails twice as fast as rfl.

Equations

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

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

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def CategoryTheory.aesop_cat? : Lean.ParserDescr

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

Equations

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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

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class CategoryTheory.Category (obj : Type u) extends CategoryTheory.CategoryStruct.{v, u} obj : 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.)

• 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) : CategoryStruct.comp (CategoryStruct.id X) f = f

  Identity morphisms are left identities for composition.

• comp_id {X Y : obj} (f : X ⟶ Y) : CategoryStruct.comp f (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) : CategoryStruct.comp (CategoryStruct.comp f g) h = CategoryStruct.comp f (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.6, ?u.6 + 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.6, ?u.6} C

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

postcompose an equation between morphisms by another morphism

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

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

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

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

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

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

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

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

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

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

class CategoryTheory.Epi {C : Type u} [Category.{v, u} C] {X 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.

• left_cancellation {Z : C} (g h : Y ⟶ Z) : CategoryStruct.comp f g = 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} [Category.{v, u} C] {X 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.

• right_cancellation {Z : C} (g h : Z ⟶ X) : CategoryStruct.comp g f = 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} [Category.{v, u} C] (X : C) : Epi (CategoryStruct.id X)

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

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

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

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

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

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

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

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

The composition of epimorphisms is again an epimorphism. This version takes Epi f and Epi g as typeclass arguments. For a version taking them as explicit arguments, see epi_comp'.

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

The composition of epimorphisms is again an epimorphism. This version takes Epi f and Epi g as explicit arguments. For a version taking them as typeclass arguments, see epi_comp.

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

The composition of monomorphisms is again a monomorphism. This version takes Mono f and Mono g as typeclass arguments. For a version taking them as explicit arguments, see mono_comp'.

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

The composition of monomorphisms is again a monomorphism. This version takes Mono f and Mono g as explicit arguments. For a version taking them as typeclass arguments, see mono_comp.

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

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

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

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

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

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

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

Equations

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