diff options
| author | Jacques Comeaux <jacquesrcomeaux@protonmail.com> | 2026-09-03 19:22:21 -0500 |
|---|---|---|
| committer | Jacques Comeaux <jacquesrcomeaux@protonmail.com> | 2026-09-03 19:22:21 -0500 |
| commit | 4e6acbf8b8084bea12327683aceb300a9b64c367 (patch) | |
| tree | 18ce77db537b4fdd7ef5be66b09a84ea22195f10 /Object/Monoid/Frobenius.agda | |
| parent | 42ab1c0b0fc5e7f16123b628dc65a38274d22f3d (diff) | |
Add Frobenius monoids and hypergraph categoriesmain
Diffstat (limited to 'Object/Monoid/Frobenius.agda')
| -rw-r--r-- | Object/Monoid/Frobenius.agda | 40 |
1 files changed, 40 insertions, 0 deletions
diff --git a/Object/Monoid/Frobenius.agda b/Object/Monoid/Frobenius.agda new file mode 100644 index 0000000..672e65e --- /dev/null +++ b/Object/Monoid/Frobenius.agda @@ -0,0 +1,40 @@ +{-# OPTIONS --without-K --safe #-} + +open import Categories.Category using (Category) +open import Categories.Category.Monoidal using (Monoidal) +open import Categories.Category.Monoidal.Symmetric using (Symmetric) +open import Level using (Level; _⊔_) + +module Object.Monoid.Frobenius {o ℓ e : Level} {C : Category o ℓ e} {M : Monoidal C} (S : Symmetric M) where + +open import Categories.Category.Monoidal.Symmetric.Properties S using () renaming (symmetric-Op to Sᵒᵖ) +open import Object.Monoid.Commutative using (IsCommutativeMonoid) + +open Category C + +record IsSpecialCommutativeFrobeniusMonoid (A : Obj) : Set (ℓ ⊔ e) where + + field + commutativeMonoid : IsCommutativeMonoid S A + cocommutativeComonoid : IsCommutativeMonoid Sᵒᵖ A + + open IsCommutativeMonoid commutativeMonoid + using (μ; η; commutative) + renaming (assoc to μ-assoc; identityˡ to μ-η-identityˡ; identityʳ to μ-η-identityʳ) + public + + open IsCommutativeMonoid cocommutativeComonoid + using () + renaming (μ to δ; η to ϵ; assoc to δ-assoc; identityˡ to δ-ϵ-identityˡ; identityʳ to δ-ϵ-identityʳ; commutative to cocommutative) + public + + field + special : μ ∘ δ ≈ id + +record SpecialCommutativeFrobeniusMonoid : Set (o ⊔ ℓ ⊔ e) where + + field + Carrier : Obj + isSpecialCommutativeFrobeniusMonoid : IsSpecialCommutativeFrobeniusMonoid Carrier + + open IsSpecialCommutativeFrobeniusMonoid isSpecialCommutativeFrobeniusMonoid public |
