feat: def of graphical species

HepLean/SpaceTime/LorentzTensor/Basic.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzTensor.GraphicalSpecies
+import HepLean.SpaceTime.LorentzVector.Basic
+/-!
+
+# Lorentz Tensors
+
+This file is currently a work-in-progress.
+
+The aim is to define Lorentz tensors, and devlop a systematic way to manipulate them.
+
+To manipulate them we will use the theory of modular operads
+(see e.g. [Raynor][raynor2021graphical]).
+
+
+-/
+
+/-- A Lorentz Tensor defined by its coordinate map. -/
+def LorentzTensor (d : ℕ) (X : FintypeCat) : Type :=
+  (X → Fin 1 ⊕ Fin d) → ℝ
+
+/-- An instance of a additive commutative monoid on `LorentzTensor`. -/
+instance (d : ℕ) (X : FintypeCat)  : AddCommMonoid (LorentzTensor d X) := Pi.addCommMonoid
+
+/-- An instance of a module on `LorentzVector`. -/
+noncomputable instance (d : ℕ) (X : FintypeCat)  : Module ℝ (LorentzTensor d X) := Pi.module _ _ _
+
+namespace LorentzTensor
+open BigOperators
+open elGr
+open CategoryTheory
+
+variable {d : ℕ} {X Y : FintypeCat}
+
+/-- The map taking a list of `LorentzVector d` indexed by `X` to a ` LorentzTensor d X`. -/
+def tmul (t : X → LorentzVector d) : LorentzTensor d X :=
+  fun f => ∏ x, (t x) (f x)
+
+/- An equivalence between `X → Fin 1 ⊕ Fin d` and `Y → Fin 1 ⊕ Fin d` given an isomorphism
+  between `X` and `Y`. -/
+def indexEquivOfIndexHom (f : X ≅ Y) : (X → Fin 1 ⊕ Fin d) ≃ (Y → Fin 1 ⊕ Fin d)  :=
+  Equiv.piCongrLeft' _ (FintypeCat.equivEquivIso.symm f)
+
+/-- Given an isomorphism of indexing sets, a linear equivalence on Lorentz tensors. -/
+noncomputable def mapOfIndexHom (f : X ≅ Y) :  LorentzTensor d Y ≃ₗ[ℝ] LorentzTensor d X :=
+  LinearEquiv.piCongrLeft' ℝ _ (indexEquivOfIndexHom f).symm
+
+/-!
+
+## Graphical species and Lorentz tensors
+
+-/
+
+/-- The graphical species defined by Lorentz tensors.
+
+For this simple case, 𝓣 gets mapped to `PUnit`, if one wishes to include fermions etc,
+then `PUnit` will change to account for the colouring of edges. -/
+noncomputable def graphicalSpecies (d : ℕ) : GraphicalSpecies where
+  obj x :=
+    match x with
+    | ⟨𝓣⟩ => PUnit
+    | ⟨as f⟩ => LorentzTensor d f
+  map {x y} f :=
+    match x, y, f with
+    | ⟨𝓣⟩, ⟨𝓣⟩, _ => 𝟙 PUnit
+    | ⟨𝓣⟩, ⟨as x⟩, ⟨f⟩ => Empty.elim f
+    | ⟨as f⟩, ⟨𝓣⟩, _ => fun _ => PUnit.unit
+    | ⟨as f⟩, ⟨as g⟩, ⟨h⟩ => (mapOfIndexHom h).toEquiv.toFun
+  map_id X := by
+    match X with
+    | ⟨𝓣⟩ => rfl
+    | ⟨as f⟩ => rfl
+  map_comp {x y z} f g := by
+    match x, y, z, f, g with
+    | ⟨𝓣⟩, ⟨𝓣⟩, ⟨𝓣⟩, _, _ => rfl
+    | _, ⟨𝓣⟩, ⟨as _⟩, _, ⟨g⟩ => exact Empty.elim g
+    | ⟨𝓣⟩, ⟨as _⟩, _, ⟨f⟩, _ =>  exact Empty.elim f
+    | ⟨as x⟩, ⟨as y⟩, ⟨as z⟩, ⟨f⟩, ⟨g⟩ => rfl
+    | ⟨as x⟩, ⟨𝓣⟩, ⟨𝓣⟩, _, _ => rfl
+    | ⟨as x⟩, ⟨as y⟩, ⟨𝓣⟩, _, _ => rfl
+
+end LorentzTensor