Discrete categories

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-05-06.
Last modified on 2023-06-25.

module category-theory.discrete-categories where
Imports 
open import category-theory.precategories

open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

Discrete precategories

Any set induces a discrete category whose objects are elements of the set and which contains no-nonidentity morphisms.

module _
  {l : Level} (X : Set l)
  where

  discrete-precategory-Set : Precategory l l
  pr1 discrete-precategory-Set = type-Set X
  pr1 (pr2 discrete-precategory-Set) x y =
    set-Prop (x ＝ y , is-set-type-Set X x y)
  pr1 (pr1 (pr2 (pr2 discrete-precategory-Set))) = concat' _
  pr2 (pr1 (pr2 (pr2 discrete-precategory-Set))) refl refl refl = refl
  pr1 (pr2 (pr2 (pr2 discrete-precategory-Set))) x = refl
  pr1 (pr2 (pr2 (pr2 (pr2 discrete-precategory-Set)))) _ = right-unit
  pr2 (pr2 (pr2 (pr2 (pr2 discrete-precategory-Set)))) _ = left-unit

Recent changes