Representable functors between categories

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-09-18.

module category-theory.representable-functors-categories where

Imports
open import category-theory.categories
open import category-theory.functors-categories
open import category-theory.natural-transformations-functors-categories
open import category-theory.representable-functors-precategories

open import foundation.category-of-sets
open import foundation.universe-levels


Idea

Given a category C and an object c, there is a functor from C to the category of sets represented by c that:

• sends an object x of C to the set hom c x and
• sends a morphism g : hom x y of C to the function hom c x → hom c y defined by postcomposition with g.

The functoriality axioms follow, by function extensionality, from associativity and the left unit law for the category C.

Definition

representable-functor-Category :
{l1 l2 : Level} (C : Category l1 l2) (c : obj-Category C) →
functor-Category C (Set-Category l2)
representable-functor-Category C =
representable-functor-Precategory (precategory-Category C)


Natural transformations between representable functors

A morphism f : hom b c in a category C defines a natural transformation from the functor represented by c to the functor represented by b. Its components hom c x → hom b x are defined by precomposition with f.

representable-natural-transformation-Category :
{l1 l2 : Level} (C : Category l1 l2) {b c : obj-Category C}
(f : hom-Category C b c) →
natural-transformation-Category
( C)
( Set-Category l2)
( representable-functor-Category C c)
( representable-functor-Category C b)
representable-natural-transformation-Category C =
representable-natural-transformation-Precategory (precategory-Category C)