Library HoTT.Categories.Limits.Functors

(co)limits assemble into functors

Require Import Category.Core Functor.Core.
Require Import KanExtensions.Functors.
Require Import Limits.Core.
Require Import FunctorCategory.Core.
Require Import Adjoint.Core.
Require Import NatCategory.

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

(co)limits assemble into functors

Local Open Scope category_scope.

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

Colimit functor, which is left adjoint to Δ

  Section colimit.
    Context `(has_colimits
              : ∀ F : Functor D C,
                  @IsColimit _ C D F (colimits F)).

TODO(jgross): We'll probably want to compose this with the functor from 1 → C to C, and then compose the adjunctions similarly. This will require turning the equality in the exponential laws into an adjunction. Probably not very hard.

    Definition colimit_functor : Functor (D → C) (1 → C)
      := left_kan_extension_functor has_colimits.

    Definition colimit_adjunction
    : colimit_functor -| diagonal_functor _ _
      := left_kan_extension_adjunction has_colimits.
  End colimit.

  Section limit.
    Context `(has_limits
              : ∀ F : Functor D C,
                  @IsLimit _ C D F (limits F)).

TODO(jgross): We'll probably want to compose this with the functor from 1 → C to C, and then compose the adjunctions similarly. This will require turning the equality in the exponential laws into an adjunction. Probably not very hard.

Limit functor, which is right adjoint to Δ

    Definition limit_functor : Functor (D → C) (1 → C)
      := right_kan_extension_functor has_limits.

    Definition limit_adjunction
    : diagonal_functor _ _ -| limit_functor
      := right_kan_extension_adjunction has_limits.
  End limit.
End functors.