module Cat.Abelian.Instances.Ab {β} where
open is-additive.Coequaliser open is-additive.Terminal open is-pre-abelian open Ab-category open is-additive open make-group
The category of abelian groupsπ
The prototypal β representative, even β example of an -enriched, and an abelian category at that, is the category of abelian groups, . For abstractly-nonsensical reasons, we could say is -enriched by virtue of being monoidal closed, but we have a concrete construction at hand: Ab-ab-category
Let us show it is additive. The terminal group is given by the terminal set, equipped with its unique group structure, and we have already computed products β they are given by direct sums.
Ab-is-additive : is-additive (Ab β) Ab-is-additive .has-ab = Ab-ab-category Ab-is-additive .has-terminal .top = from-commutative-group (Zero-group {β}) (Ξ» x y β refl) Ab-is-additive .has-terminal .hasβ€ x = contr (total-hom (Ξ» _ β lift tt) (record { pres-β = Ξ» x y i β lift tt })) Ξ» x β Homomorphism-path Ξ» _ β refl Ab-is-additive .has-prods A B .Product.apex = A β B Ab-is-additive .has-prods A B .Product.Οβ = _ Ab-is-additive .has-prods A B .Product.Οβ = _ Ab-is-additive .has-prods A B .Product.has-is-product = Direct-sum-is-product