The precategory of semigroups

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2022-03-17.
Last modified on 2023-11-27.

module group-theory.precategory-of-semigroups where
Imports
open import category-theory.large-precategories

open import foundation.universe-levels

open import group-theory.homomorphisms-semigroups
open import group-theory.semigroups

Idea

Semigroups and semigroup homomorphisms form a precategory.

Definition

The precategory of semigroups

instance
  Semigroup-Large-Precategory : Large-Precategory lsuc (_⊔_)
  obj-Large-Precategory Semigroup-Large-Precategory = Semigroup
  hom-set-Large-Precategory Semigroup-Large-Precategory = hom-set-Semigroup
  comp-hom-Large-Precategory Semigroup-Large-Precategory
    {X = G} {H} {K} =
    comp-hom-Semigroup G H K
  id-hom-Large-Precategory Semigroup-Large-Precategory
    {X = G} =
    id-hom-Semigroup G
  associative-comp-hom-Large-Precategory Semigroup-Large-Precategory
    {X = G} {H} {K} {L} =
    associative-comp-hom-Semigroup G H K L
  inv-associative-comp-hom-Large-Precategory Semigroup-Large-Precategory
    {X = G} {H} {K} {L} =
    inv-associative-comp-hom-Semigroup G H K L
  left-unit-law-comp-hom-Large-Precategory Semigroup-Large-Precategory
    {X = G} {H} =
    left-unit-law-comp-hom-Semigroup G H
  right-unit-law-comp-hom-Large-Precategory Semigroup-Large-Precategory
    {X = G} {H} =
    right-unit-law-comp-hom-Semigroup G H

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