Library HoTT.categories.Limits.Core

Limits and Colimits

Require Import Category.Core Functor.Core NaturalTransformation.Core.
Require Import Functor.Composition.Core.
Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core.
Require Import UniversalProperties KanExtensions.Core InitialTerminalCategory.Core NatCategory.
Require Import Functor.Paths NaturalTransformation.Paths.
Require Import Comma.Core.
Require Import Equivalences.

Set Universe Polymorphism.
Set Implicit Arguments.
Generalizable All Variables.
Set Asymmetric Patterns.

Local Open Scope functor_scope.
Local Open Scope category_scope.

The diagonal or "constant diagram" functor Δ

Section diagonal_functor.
  Context `{Funext}.
  Variables C D : PreCategory.

Quoting Dwyer and Spalinski:
There is a diagonal or ``constant diagram'' functor
     Δ : C → Cᴰ,
which carries an object X : C to the constant functor Δ X : D C (by definition, this ``constant functor'' sends each object of D to X and each morphism of D to Identity X). The functor Δ assigns to each morphism f : X X' of C the constant natural transformation t(f) : Δ X Δ X' determined by the formula t(f) d = f for each object d of D.
We use rather than C for judgemental compatibility with Kan extensions.

  Definition diagonal_functor : Functor (1 C) (D C)
    := @pullback_along _ D 1 C (Functors.to_terminal _).

  Definition diagonal_functor' : Functor C (D C)
    := diagonal_functor o ExponentialLaws.Law1.Functors.inverse _.

  Section convert.
    Lemma diagonal_functor_diagonal_functor' X
    : diagonal_functor X = diagonal_functor' (X (center _)).
    Proof.
      path_functor.
      simpl.
      repeat (apply path_forall || intro).
      apply identity_of.
    Qed.
  End convert.
End diagonal_functor.

Arguments diagonal_functor : simpl never.

Section diagonal_functor_lemmas.
  Context `{Funext}.
  Variables C D D' : PreCategory.

  Lemma compose_diagonal_functor x (F : Functor D' D)
  : diagonal_functor C D x o F = diagonal_functor _ _ x.
  Proof.
    path_functor.
  Qed.

  Definition compose_diagonal_functor_natural_transformation
             x (F : Functor D' D)
  : NaturalTransformation (diagonal_functor C D x o F) (diagonal_functor _ _ x)
    := Build_NaturalTransformation
         (diagonal_functor C D x o F)
         (diagonal_functor _ _ x)
         (fun zidentity _)
         (fun _ _ _transitivity
                         (left_identity _ _ _ _)
                         (symmetry
                            _ _
                            (right_identity _ _ _ _))).
End diagonal_functor_lemmas.

Hint Rewrite @compose_diagonal_functor : category.

Section Limit.
  Context `{Funext}.
  Variables C D : PreCategory.
  Variable F : Functor D C.

Definition of Limit

Quoting Dwyer and Spalinski:
Let D be a small category and F : D C a functor. A limit for F is an object L of C together with a natural transformation t : Δ L F such that for every object X of C and every natural transformation s : Δ X F, there exists a unique map s' : X L in C such that t (Δ s') = s.

  Definition IsLimit
    := @IsRightKanExtensionAlong _ D 1 C (Functors.to_terminal _) F.

Quoting Dwyer and Spalinski:
Let D be a small category and F : D C a functor. A colimit for F is an object c of C together with a natural transformation t : F Δ c such that for every object X of C and every natural transformation s : F Δ X, there exists a unique map s' : c X in C such that (Δ s') t = s.

Definition of Colimit

  Definition IsColimit
    := @IsLeftKanExtensionAlong _ D 1 C (Functors.to_terminal _) F.
TODO(JasonGross): Figure out how to get good introduction and elimination rules working, which don't mention spurious identities.
End Limit.

TODO(JasonGross): Port MorphismsBetweenLimits from catdb