SPECTRA TYPE

The type models spectra in stable homotopy theory, sequences of types with structure maps , used in K-theory, cohomology, and quantum field theory. Spectra capture stable invariants.

In the cohesive type system, is a parameterless type in , not a universe, supporting operations like suspension , wedge sums , homomorphisms , and qubits .

Formation

(-Formation). The type is a base type in , representing all spectra, stable under suspension.

def Sp : U_{(0,0)}
(* Type of all spectra *)

Introduction

(-Introduction). Terms of type are introduced via a base constructor or operations like suspension (), wedge sums (), homomorphisms (), or qubits (), forming stable homotopy classes.

def sp : Sp
def susp (E : Sp) : Sp
def wedge (E F : Sp) : Sp
def hom_spec (E F : Sp) : Sp
def qubit (C H : U_{(0,0)}) : Sp
(* Constructors for spectra *)

Elimination

(-Elimination). The eliminator maps terms to properties in , analyzing stable homotopy structure.

def sp_ind (beta : Sp -> U_{(0,0)}) (h : Π (E : Sp), beta E) : Π (E : Sp), beta E
(* Eliminator for spectra *)

Theorems

(Suspension Invariance). For , , preserving stable homotopy structure.

def susp_inv (E : Sp) : Sp

(Wedge Sum Closure). For , , the coproduct in spectra.

def wedge_closure (E F : Sp) : Sp

(Homomorphism Spectrum). For , , the spectrum of stable maps.

def hom_closure (E F : Sp) : Sp

(Qubit as Spectrum). For , , connecting quantum states to spectra.

def qubit_spectrum (C H : U_{(0,0)}) : Sp