From 43942e78f4b3ed4cfca3b508f06161d0983c545e Mon Sep 17 00:00:00 2001
From: Hannes Schulze <haschu0103@outlook.de>
Date: Fri, 5 Dec 2025 23:51:59 +0100
Subject: Alg category

---
 Algebra/Semantics.lean | 45 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 45 insertions(+)

diff --git a/Algebra/Semantics.lean b/Algebra/Semantics.lean
index f5160dd..13b54a0 100644
--- a/Algebra/Semantics.lean
+++ b/Algebra/Semantics.lean
@@ -2,8 +2,12 @@ import Mathlib
 import Algebra.Terms
 import Algebra.Derivations
 
+open CategoryTheory
+
 namespace NominalAlgebra
 
+section AlgebraDefinition
+
 variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
 variable (sig : Signature)
 
@@ -19,6 +23,7 @@ structure Algebra where
 instance : CoeFun (Algebra 𝔸 sig) (fun _ => ℕ → Nom 𝔸) where
   coe A := A.carrier
 
+@[ext]
 structure Algebra.Hom (A B : Algebra 𝔸 sig) where
   toFun : (n : ℕ) → (A n →ₑ[𝔸] B n)
   preserve_pure : ∀ (f : sig.pure) {m : ℕ} (args : sig.pureArgs f → A m),
@@ -31,4 +36,44 @@ structure Algebra.Hom (A B : Algebra 𝔸 sig) where
 instance {A B : Algebra 𝔸 sig} : CoeFun (Algebra.Hom 𝔸 sig A B) (fun _ => (n : ℕ) → (A n →ₑ[𝔸] B n)) where
   coe h := h.toFun
 
+end AlgebraDefinition
+
+section Algebra
+
+variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable {sig : Signature}
+
+namespace Algebra
+
+def Hom.id (A : Algebra 𝔸 sig) : Hom 𝔸 sig A A where
+  toFun _ := EquivariantMap.id 𝔸
+  preserve_pure _ _ _ := rfl
+  preserve_free _ _ _ _ := rfl
+  preserve_bound _ _ _ _ := rfl
+
+def Hom.comp {A B C : Algebra 𝔸 sig} (h : Hom 𝔸 sig B C) (g : Hom 𝔸 sig A B) : Hom 𝔸 sig A C where
+  toFun n := EquivariantMap.comp 𝔸 (h n) (g n)
+  preserve_pure f m args := by
+    rw [EquivariantMap.comp_app, g.preserve_pure, h.preserve_pure]
+    rfl
+  preserve_free a f m args := by
+    rw [EquivariantMap.comp_app, g.preserve_free, h.preserve_free]
+    rfl
+  preserve_bound a f m args := by
+    rw [EquivariantMap.comp_app, g.preserve_bound, h.preserve_bound]
+    rfl
+
+instance : Category (Algebra 𝔸 sig) where
+  Hom := Hom 𝔸 sig
+  id := Hom.id
+  comp h g := Hom.comp g h
+
+def forgetfulFunctor : Algebra 𝔸 sig ⥤ Nom 𝔸 where
+  obj X := X 0
+  map f := f.toFun 0
+
+end Algebra
+
+end Algebra
+
 end NominalAlgebra
-- 
cgit v1.3.1