DIFFERENTIAL COHOMOLOGY

The type models the space of differential -forms on a smooth set , a fundamental component of differential cohomology, which refines de Rham cohomology with differential forms. This is used in smooth geometry, gauge theory, and quantum field theory.

In Urs, is a type in , parameterized by and , supporting operations like the wedge product and exterior derivative, with applications to gauge fields in and graded forms in .

Formation

(-Formation). The type is a type in , formed for each and , representing the space of -forms .

def Forms (n : Nat) (X : SmthSet) : U_0
(* Type of n-forms Ω^n(X) *)

Introduction

(-Introduction). Terms of type are differential -forms, introduced via a constructor, with operations like the wedge product.

def form (n : Nat) (X : SmthSet) : Forms n X
(* Constructor for n-forms *)
def wedge (n m : Nat) (X : SmthSet) (α : Forms n X) (β : Forms m X) : Forms (n+m) X
(* Wedge product: α ∧ β *)
(* Differential form introduction *)

Elimination

(-Elimination). The eliminator for maps differential forms to properties in , analyzing their structure.

def forms_ind (n : Nat) (X : SmthSet)
    (phi : Forms n X -> U_0)
    (h : Π (ω : Forms n X), phi ω)
  : Π (ω : Forms n X), phi ω

Theorems

(Exterior Derivative). For , , the exterior derivative maps -forms to -forms, satisfying .

def ext_der (n : Nat) (X : SmthSet) : Forms n X -> Forms (n+1) X
def ext_der_nil (n : Nat) (X : SmthSet) (ω : Forms n X) : (ext_der (n+1) X (ext_der n X ω)) = 0
(* Exterior derivative: d : Ω^n(X) → Ω^(n+1)(X), d ∘ d = 0 *)

(Wedge Product Properties). The wedge product is graded commutative, reflecting the structure of differential forms.

def wedge_comm (n m : Nat) (X : SmthSet) (α : Forms n X) (β : Forms m X)
  : Path (Forms (n+m) X) (wedge n m X α β) ((-1)^(n * m) * (wedge m n X β α))
(* Graded commutativity: α ∧ β = (-1)^(n m) (β ∧ α) *)

(De Rham Cohomology Link). For , , contributes to de Rham cohomology via closed and exact forms.

def closed_forms (n : Nat) (X : SmthSet) (ω : Forms n X) : Prop := (ext_der n X ω) = 0
def exact_forms (n : Nat) (X : SmthSet) (ω : Forms n X) : Prop := Σ (α : Forms (n-1) X), (ext_der (n-1) X α) = ω
def de_rham (n : Nat) (X : SmthSet) : Hom_Sp (closed_forms n X, exact_forms n X)
(* De Rham cohomology: H^n_dR(X) *)

(Gauge Field Application). For , models gauge fields in quantum systems, as in .

def gauge_field (X : SmthSet) : Forms 1 X -> Forms 2 X := ext_der 1 X
def gauge_field_curv (X : SmthSet) (A : Forms 1 X) : (gauge_field X A) = (ext_der 1 X A) := refl
(* Gauge field: F = dA *)