blob: a792c6d1eb3e54f0c7798ca7bc593da2ae286524 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
|
{-# OPTIONS --without-K --safe #-}
module Category.Instance.SymMonPre where
import Category.Instance.MonoidalPreorders as MP using (_≗_; module ≗)
open import Categories.Category using (Category)
open import Categories.Category.Helper using (categoryHelper)
open import Category.Instance.MonoidalPreorders using (MonoidalPreorders)
open import Level using (Level; suc; _⊔_)
open import Preorder.Monoidal using (SymmetricMonoidalPreorder; SymmetricMonoidalMonotone)
open import Relation.Binary using (IsEquivalence)
private
identity : {c ℓ e : Level} (A : SymmetricMonoidalPreorder c ℓ e) → SymmetricMonoidalMonotone A A
identity {c} {ℓ} {e} A = record
{ monoidalMonotone = id {monoidalPreorder}
}
where
open SymmetricMonoidalPreorder A using (monoidalPreorder)
open Category (MonoidalPreorders c ℓ e) using (id)
compose
: {c ℓ e : Level}
{P Q R : SymmetricMonoidalPreorder c ℓ e}
→ SymmetricMonoidalMonotone Q R
→ SymmetricMonoidalMonotone P Q
→ SymmetricMonoidalMonotone P R
compose {c} {ℓ} {e} {R = R} G F = record
{ monoidalMonotone = G.monoidalMonotone ∘ F.monoidalMonotone
}
where
module G = SymmetricMonoidalMonotone G
module F = SymmetricMonoidalMonotone F
open Category (MonoidalPreorders c ℓ e) using (_∘_)
module _ {c₁ c₂ ℓ₁ ℓ₂ e₁ e₂ : Level} {A : SymmetricMonoidalPreorder c₁ ℓ₁ e₁} {B : SymmetricMonoidalPreorder c₂ ℓ₂ e₂} where
open SymmetricMonoidalMonotone using () renaming (monoidalMonotone to mM)
-- Pointwise equality of symmetric monoidal monotone maps
_≗_ : (f g : SymmetricMonoidalMonotone A B) → Set (c₁ ⊔ ℓ₂)
f ≗ g = mM f MP.≗ mM g
infix 4 _≗_
≗-isEquivalence : IsEquivalence _≗_
≗-isEquivalence = let open MP.≗ in record
{ refl = λ {x} → refl {x = mM x}
; sym = λ {f g} → sym {x = mM f} {y = mM g}
; trans = λ {f g h} → trans {i = mM f} {j = mM g} {k = mM h}
}
module ≗ = IsEquivalence ≗-isEquivalence
-- The category of symmetric monoidal preorders
SymMonPre : (c ℓ e : Level) → Category (suc (c ⊔ ℓ ⊔ e)) (c ⊔ ℓ ⊔ e) (c ⊔ ℓ)
SymMonPre c ℓ e = categoryHelper record
{ Obj = SymmetricMonoidalPreorder c ℓ e
; _⇒_ = SymmetricMonoidalMonotone
; _≈_ = _≗_
; id = λ {A} → identity A
; _∘_ = λ {A B C} f g → compose f g
; assoc = λ {_ _ _ D _ _ _ _} → Eq.refl D
; identityˡ = λ {_ B _ _} → Eq.refl B
; identityʳ = λ {_ B _ _} → Eq.refl B
; equiv = ≗-isEquivalence
; ∘-resp-≈ = λ {_ _ C _ h} f≈h g≈i → Eq.trans C f≈h (cong h g≈i)
}
where
open SymmetricMonoidalMonotone using (cong)
open SymmetricMonoidalPreorder using (refl; module Eq)
|
