GENERALIZED K-THEORY TYPE

The type represents generalized K-theory, a topological invariant used to classify vector bundles or operator algebras over a space, twisted by a groupoid. It is a cornerstone of algebraic topology and mathematical physics, with applications in quantum field theory, string theory, and index theory.

In the cohesive type system, operates on smooth sets and groupoids , producing a type in the universe . It incorporates a twist to account for non-trivial topological structures, making it versatile for modeling complex physical systems.

Formation

(-Formation). The generalized K-theory type is formed over a term , a groupoid , and a twist , yielding a type in the universe .

type exp =
  | KU^G of exp * exp * exp  (* KU^G X G τ *)

Introduction

(-Introduction). A term of type is introduced by constructing a generalized K-theory class, representing a stable equivalence class of vector bundles or operators over , twisted by and .

let KU^G_intro (x : exp) (g : exp) (tau : exp) : exp =
    KU^G (x, g, tau)

Elimination

(-Elimination). The eliminator for allows reasoning about generalized K-theory classes by mapping them to properties or types dependent on , typically by analyzing the underlying bundle or operator structure over .

let KU^G_ind (x : exp) (g : exp) (tau : exp) (beta : exp) (h : exp) : exp =
    (* Hypothetical eliminator *)
    App (Var "KU^GInd", KU^G (x, g, tau))

Theorems

(K-Theory Stability). The type is stable under suspension, meaning it is invariant under the suspension operation in the spectrum, reflecting its role in stable homotopy theory.

let KU^G_stability (x : exp) (g : exp) (tau : exp) : exp =
    Path (Universe (0, Bose),
          KU^G (x, g, tau),
          KU^G (Susp x, g, tau))
    (* Path type represents equality *)

(Refinement to Differential K-Theory). The type can be refined to differential K-theory by incorporating a connection, as provided by .

let KU^G_to_KU_\flat^G (x : exp) (g : exp) (tau : exp) (conn : exp) : exp =
    KU_\flat^G (x, g, tau, conn)