Documentation

Mathlib.CategoryTheory.Category.Basic

Categories #

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

Notations #

Introduces notations in the CategoryTheory scope

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

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

Porting note #

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

A wrapper for ext that we can pass to aesop.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    class CategoryTheory.CategoryStruct (obj : Type u) extends Quiver :
    Type (max u (v + 1))
    • Hom : objobjType 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.

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

    Instances

      Notation for the identity morphism in a category.

      Equations
      Instances For

        Notation for composition of morphisms in a category.

        Equations
        Instances For

          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.
          Instances For

            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.
            Instances For

              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.
              Instances For
                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.

                Instances
                  @[inline, reducible]
                  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
                  Instances For
                    @[inline, reducible]
                    abbrev CategoryTheory.SmallCategory (C : Type u) :
                    Type (u + 1)

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

                    Equations
                    Instances For
                      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) :

                      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) :

                      precompose an equation between morphisms by another morphism

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

                        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
                        • One or more equations did not get rendered due to their size.
                        Instances For
                          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} h => CategoryTheory.CategoryStruct.comp f h) = fun {Z} 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 : (fun {X} h => CategoryTheory.CategoryStruct.comp h f) = fun {X} h => CategoryTheory.CategoryStruct.comp h g) :
                          f = g
                          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) :
                          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) :
                          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) :

                          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.

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

                            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.

                            Instances
                              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) :
                              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) :