cat/src/Cat/Category/Free.agda

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module Cat.Category.Free where
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open import Agda.Primitive
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open import Cubical hiding (Path)
open import Data.Product
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open import Cat.Category as C
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module _ { ' : Level} ( : Category ') where
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private
open module = Category
Obj = .Object
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postulate
Path : ( a b : Obj ) Set '
emptyPath : (o : Obj) Path o o
concatenate : {a b c : Obj} Path b c Path a b Path a c
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private
module _ {A B C D : Obj} {r : Path A B} {q : Path B C} {p : Path C D} where
postulate
p-assoc : concatenate {A} {C} {D} p (concatenate {A} {B} {C} q r)
concatenate {A} {B} {D} (concatenate {B} {C} {D} p q) r
module _ {A B : Obj} {p : Path A B} where
postulate
ident-r : concatenate {A} {A} {B} p (emptyPath A) p
ident-l : concatenate {A} {B} {B} (emptyPath B) p p
Free : Category '
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Free = record
{ Object = Obj
; Arrow = Path
; 𝟙 = λ {o} emptyPath o
; _⊕_ = λ {a b c} concatenate {a} {b} {c}
; isCategory = record { assoc = p-assoc ; ident = ident-r , ident-l }
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}