feat: Add Feynman diagram_start

HepLean/FeynmanDiagrams/Basic.lean

@@ -0,0 +1,95 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import Mathlib.Logic.Equiv.Fin
+import Mathlib.Tactic.FinCases
+import Mathlib.Data.Finset.Card
+import Mathlib.CategoryTheory.IsomorphismClasses
+import LeanCopilot
+/-!
+# Feynman diagrams in Phi^4 theory
+
+The aim of this file is to start building up the theory of Feynman diagrams in the context of
+Phi^4 theory.
+
+-/
+
+namespace PhiFour
+
+/-- The type of vacuum Feynman diagrams for Phi-4 theory. -/
+structure VacuumFeynmanDiagram where
+  HalfEdge : Type
+  Edge : Type
+  Vertex : Type
+  𝓔 : HalfEdge → Edge
+  𝓥 : HalfEdge → Vertex
+  𝓔Fiber : ∀ x,  CategoryTheory.IsIsomorphic (𝓔 ⁻¹' {x} : Type) (Fin 2)
+  𝓥Fiber : ∀ x,  CategoryTheory.IsIsomorphic (𝓥 ⁻¹' {x} : Type) (Fin 3)
+
+
+
+namespace VacuumFeynmanDiagram
+
+open CategoryTheory
+
+variable (F G : VacuumFeynmanDiagram)
+
+/-- The definition of a morphism between two `VacuumFeynmanDiagram`. -/
+structure Hom where
+  halfEdgeMap : F.HalfEdge ⟶ G.HalfEdge
+  edgeMap : F.Edge ⟶ G.Edge
+  vertexMap : F.Vertex ⟶ G.Vertex
+
+namespace Hom
+
+/-- The identity morphism for an object in `VacuumFeynmanDiagram`. -/
+def id : Hom F F where
+  halfEdgeMap := 𝟙 F.HalfEdge
+  edgeMap := 𝟙 F.Edge
+  vertexMap := 𝟙 F.Vertex
+
+/-- The composition of morphisms between two `VacuumFeynmanDiagram`. -/
+def comp {F G H : VacuumFeynmanDiagram} (f : Hom F G) (g : Hom G H) : Hom F H where
+  halfEdgeMap := f.halfEdgeMap ≫ g.halfEdgeMap
+  edgeMap := f.edgeMap ≫ g.edgeMap
+  vertexMap := f.vertexMap ≫ g.vertexMap
+
+end Hom
+
+/-- `VacuumFeynmanDiagram` has the structure of a category. -/
+instance : Category VacuumFeynmanDiagram where
+  Hom := Hom
+  id := Hom.id
+  comp := Hom.comp
+
+/-- The functor from the category `VacuumFeynmanDiagram` to `Type` defined by
+  mapping to the `Type` of half edges.  -/
+def toHalfEdge : VacuumFeynmanDiagram ⥤ Type where
+  obj F := F.HalfEdge
+  map f := f.halfEdgeMap
+
+@[simp]
+lemma 𝓔_preimage_nonempty (x : F.Edge) : (F.𝓔 ⁻¹' {x}).Nonempty := by
+  obtain ⟨_, f, _, _⟩ := F.𝓔Fiber x
+  exact ⟨f 0,  Subtype.coe_prop (f 0)⟩
+
+@[simp]
+lemma 𝓥_preimage_nonempty (x : F.Vertex) : (F.𝓥 ⁻¹' {x}).Nonempty := by
+  obtain ⟨_, f, _, _⟩ := F.𝓥Fiber x
+  exact ⟨f 0,  Subtype.coe_prop (f 0)⟩
+
+@[simp]
+lemma 𝓔_surjective : Function.Surjective F.𝓔 := by
+  exact fun x => F.𝓔_preimage_nonempty x
+
+@[simp]
+lemma 𝓥_surjective : Function.Surjective F.𝓥 := by
+  exact fun x => F.𝓥_preimage_nonempty x
+
+
+end VacuumFeynmanDiagram
+
+
+end PhiFour