{"id":25882072,"url":"https://github.com/groupoid/urs","last_synced_at":"2026-03-01T03:03:41.003Z","repository":{"id":279873396,"uuid":"940272742","full_name":"groupoid/urs","owner":"groupoid","description":"🧊 Еквіваріантна теорія типів супергеометрії","archived":false,"fork":false,"pushed_at":"2025-05-14T00:19:37.000Z","size":4422,"stargazers_count":2,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-10-29T00:39:21.201Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":"http://urs.groupoid.space/","language":"Pug","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/groupoid.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null}},"created_at":"2025-02-27T22:31:04.000Z","updated_at":"2025-05-14T00:19:40.000Z","dependencies_parsed_at":null,"dependency_job_id":"279c959f-3484-4e8f-b989-c061034e6dd3","html_url":"https://github.com/groupoid/urs","commit_stats":null,"previous_names":["groupoid/urs"],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/groupoid/urs","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/groupoid%2Furs","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/groupoid%2Furs/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/groupoid%2Furs/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/groupoid%2Furs/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/groupoid","download_url":"https://codeload.github.com/groupoid/urs/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/groupoid%2Furs/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":286080680,"owners_count":29959284,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2026-03-01T01:47:18.291Z","status":"online","status_checked_at":"2026-03-01T02:00:07.437Z","response_time":124,"last_error":null,"robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2025-03-02T15:57:07.838Z","updated_at":"2026-03-01T03:03:40.994Z","avatar_url":"https://github.com/groupoid.png","language":"Pug","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Urs: Equivariant Super HoTT\n\n## Abstract\n\nWe present a layered type theory that integrates three foundational frameworks:\nHomotopy Type System (HTS), de Rham Cohesive Modal Type Theory (CMTT), and Equivariant Super Type Theory (ESTT).\nThis system builds a progressive structure for formalizing\nmathematical and physical concepts, from homotopy and higher categorical\nstructures, through geometric cohesion and differential properties,\nto the rich graded and equivariant world of supergeometry.\nInspired by Urs Schreiber’s work on equivariant super homotopy theory,\nthis layered approach offers a modular, type-theoretic foundation for\nsynthetic supergeometry and beyond.\n\nType theory has emerged as a powerful language for mathematics and physics,\nunifying computation, logic, and structure. This article introduces\na three-layered type theory that extends Martin-Löf’s intensional\ntype theory into a framework capable of capturing homotopy, cohesion, and supergeometry:\n\n* Homotopy Type System (HTS): foundation for higher categorical structures via types as ∞-groupoids.\n* Cohesive Modal Type Theory: modal operators for geometric cohesion and differential structure.\n* Equivariant Super Type Theory (ESTT): Thegraded universes/tensors, group actions, and super-modality.\n\nEach layer builds on the previous, culminating in a system tailored to\nformalize superpoints `(ℝᵐ|ⁿ)`, supersymmetry, and equivariant structures,\nas exemplified in Schreiber’s \"Equivariant Super Homotopy Theory\" (2019).\n\n## Syntax\n\n```OCaml\ntype grade = Bose | Fermi (* Universe grades: bosonic (0) or fermionic (1) *)\n\ntype exp =\n  (* MLTT/HoTT Core *)\n  | Universe of int * grade           (* U i g *)\n  | Var of string                     (* x *)\n  | Forall of string * exp * exp      (* Π(x:A).B *)\n  | Lam of string * exp * exp         (* λx:A. b *)\n  | App of exp * exp                  (* f a *)\n  | Path of exp * exp * exp           (* Id_A(u,v) *)\n  | Transport of exp * exp * exp      (* transport A p t *)\n\n  (* Cohesive Types *)\n  | SmthSet                           (* SmthSet *)\n  | Plot of int * exp * exp           (* plt n X φ *)\n  | Flat of exp                       (* ♭ A *)\n  | Sharp of exp                      (* ♯ A *)\n  | Shape of exp                      (* ℑ A *)\n  | Bosonic of exp                    (* ◯ A *)\n\n  (* Graded and Supergeometry *)\n  | Tensor of exp * exp               (* A ⊗ B *)\n  | SupSmthSet                        (* SupSmthSet *)\n\n  (* Groupoids *)\n  | Grpd of int                       (* Grpd n *)\n  | Comp of int * exp * exp * exp     (* comp n G a b *)\n\n  (* Spectra and Stable Homotopy *)\n  | Spectrum                          (* Spectrum *)\n  | Susp of exp                       (* Susp A *)\n  | Wedge of exp * exp                (* A ∧ B *)\n  | HomSpec of exp * exp              (* [A, B] *)\n\n  (* TED-K *)\n  | KU_G of exp * exp * exp           (* KU_G^τ(X; τ) *)\n  | Qubit of exp * exp                (* [Config, Fred^0 H] *)\n  | Config of int * exp               (* Config^n(X) *)\n  | Braid of int * exp                (* BB_n *)\n\n  (* Differential Cohomology *)\n  | Forms of int * exp                (* Ω^n(X) *)\n  | Diff of int * exp                 (* d ω *)\n  | DiffKU_G of exp * exp * exp * exp (* DiffKU_G^τ(X; τ, conn) *)\n```\n\n## Semantics\n\n### Formation\n\n* Graded Universes: ⊢ Uᵢᴵ⁰ᴵ : Uᵢ₊₁ᴵ⁰ᴵ, ⊢ Uᵢᴵ¹ᴵ : Uᵢ₊₁ᴵ⁰ᴵ.\n* Graded Tensor: Γ ⊢ A : Uᵢ^g₁, Γ ⊢ B : Uᵢ^g₂ → Γ ⊢ A ⊗ B : Uᵢ^(g₁ + g₂) Γ ⊢ a : A, Γ ⊢ b : B → Γ ⊢ a ⊗ b : A ⊗ B Γ ⊢ a : A^g₁, Γ ⊢ b : B^g₂ → Γ ⊢ a ⊗ b = (−1)^(g₁ g₂) b ⊗ a : A ⊗ B.\n* Group Action: Γ, g : 𝔾 ⊢ A : Uᵢ^g → Γ ⊢ 𝔾 → A : Uᵢ^g.\n* Super Type Theory: Uᵍᵢ`|` 𝖘 A `|` 𝔾 → A.\n* Super Modality: Γ ⊢ A : Uᵢ^g → Γ ⊢ 𝖘 A : Uᵢ^g.\n* Cohesive Type Theory: ♭ `|` ♯ `|` ℑ `|` ◯ (four built-in modalities).\n* Flat Codiscrete: Γ ⊢ A : Uᵢ^g → Γ ⊢ ♭ A : Uᵢ^g\n* Sharp Discrete:  Γ ⊢ A : Uᵢ^g → Γ ⊢ ♯ A : Uᵢ^g\n* Bosonic: Γ ⊢ A : Uᵢ⁽ᵍ⁾  →  Γ ⊢ ◯ A : Uᵢ⁽⁰⁾\n* Fermionic: Γ ⊢ A : Uᵢ⁽ᵍ⁾  →  Γ ⊢ ℑ A : Uᵢ⁽¹⁾\n\n### Introduction\n\n* Graded Tensor: Γ ⊢ a : A, Γ ⊢ b : B → Γ ⊢ a ⊗ b : A ⊗ B.\n* Group Action: Γ, g : 𝔾 ⊢ a : A → Γ ⊢ λg.a : 𝔾 → A.\n* Super Modality: Γ ⊢ a : A → Γ ⊢ 𝖘-intro(a) : 𝖘 A.\n* Bosinic: Γ ⊢ a : A  →  Γ ⊢ ◯ a : ◯ A, Γ ⊢ a : A  →  Γ ⊢ ◯ a : ◯ A,  ◯ a := η_◯ a, Γ ⊢ η_◯ : A → ◯ A, Γ ⊢ μ_◯ : ◯ (◯ A) → ◯ A,  μ_◯ ≃ id_◯A\n* Fermionic: Γ ⊢ a : A  →  Γ ⊢ ℑ a : ℑ A, Γ ⊢ a : A  →  Γ ⊢ ℑ a : ℑ A,  ℑ a := η_ℑ a, Γ ⊢ ε_ℑ : ℑ A → A, Γ ⊢ δ_ℑ : ℑ A → ℑ (ℑ A),  δ_ℑ ≃ id_ℑA\n\n### Elimation\n\n* Graded Tensor: Γ ⊢ t : A ⊗ B, Γ, x : A, y : B ⊢ C : Uᵢ^g → Γ ⊢ ⊗-elim(t, x.y.C) : C[fst(t)/x, snd(t)/y].\n* Group Action: Γ ⊢ t : 𝔾 → A, Γ ⊢ g : 𝔾 → Γ ⊢ t g : A.\n* Super Modality: Γ ⊢ t : 𝖘 A, Γ, x : A ⊢ B : Uᵢ^g, Γ, x : A ⊢ f : B → Γ ⊢ 𝖘-elim(t, x.B, f) : B[𝖘-intro(t)/x].\n* Bosonic: Γ ⊢ ◯ A : Uᵢ⁽⁰⁾  →  Γ ⊢ η_◯ : A → ◯ A, Γ ⊢ ◯ (◯ A) : Uᵢ⁽⁰⁾  →  Γ ⊢ μ_◯ : ◯ (◯ A) → ◯ A, μ_◯ ≃ id_◯A\n* Fermionic: Γ ⊢ ℑ A : Uᵢ⁽¹⁾  →  Γ ⊢ δ_ℑ : ℑ A → ℑ (ℑ A), δ_ℑ ≃ id_ℑA\n\n### Computation\n\n* Graded Tensor: Γ ⊢ a : A, Γ ⊢ b : B, Γ, x : A, y : B ⊢ C : Uᵢ^g → Γ ⊢ ⊗-elim(a ⊗ b, x.y.C) = C[a/x, b/y] : C[a/x, b/y].\n* Graded Commutativity: Γ ⊢ a : A^g₁, Γ ⊢ b : B^g₂ → Γ ⊢ a ⊗ b = (−1)^(g₁ g₂) b ⊗ a : A ⊗ B.\n* Group Action: Γ, g : 𝔾 ⊢ a : A, Γ ⊢ h : 𝔾 → Γ ⊢ (λg.a) h = a[h/g] : A.\n* Super Modality: Γ ⊢ a : A, Γ, x : A ⊢ B : Uᵢ^g, Γ, x : A ⊢ f : B → Γ ⊢ 𝖘-elim(𝖘-intro(a), x.B, f) = f[a/x] : B[a/x], ℑ (A ⊗ B) ≃ ℑ A ⊗ ◯ B ⊕ ◯ A ⊗ ℑ B.\n* Bosonic: Γ ⊢ a : A  →  Γ ⊢ ◯ a = η_◯ a : ◯ A, Γ ⊢ b : ◯ (◯ A)  →  Γ ⊢ μ_◯ b = b : ◯ A\n* Fermionic: Γ ⊢ a : A⁽¹⁾  →  Γ ⊢ ε_ℑ (ℑ a) = a : A, Γ ⊢ d : ℑ A  →  Γ ⊢ δ_ℑ d = d : ℑ A\n\n### Uniqueness\n\n* Graded Tensor: Γ ⊢ t : A ⊗ B, Γ ⊢ u : A ⊗ B, Γ ⊢ fst(t) = fst(u) : A, snd(t) = snd(u) : B → Γ ⊢ t = u : A ⊗ B.\n* Group Action Γ ⊢ t, u : 𝔾 → A, Γ, g : 𝔾 ⊢ t g = u g : A → Γ ⊢ t = u : 𝔾 → A.\n* Super Modality:  Γ ⊢ t,u : 𝖘 A, Γ ⊢ s-elim(t, x.A, x) = 𝖘-elim(u, x.A, x) : A → Γ ⊢ t = u : 𝖘 A.\n* Bosonic: Γ ⊢ f : ◯ A → B, Γ ⊢ g : ◯ A → B, Γ ⊢ ∀ (a : A), f (η_◯ a) = g (η_◯ a)  →  Γ ⊢ f = g : ◯ A → B\n* Fermionic: Γ ⊢ f : B → ℑ A, Γ ⊢ g : B → ℑ A, Γ ⊢ ∀ (b : B), ε_ℑ (f b) = ε_ℑ (g b)  →  Γ ⊢ f = g : B → ℑ A\n\n### Coherenses\n\nTensor and Modalities:\n\n```\nΓ ⊢ A : Uᵢ⁽ᵍ¹⁾, Γ ⊢ B : Uᵢ⁽ᵍ²⁾  →  Γ ⊢ ◯ (A ⊗ B) ≃ ◯ A ⊗ ◯ B : Uᵢ⁽⁰⁾\nΓ ⊢ A : Uᵢ⁽ᵍ¹⁾, Γ ⊢ B : Uᵢ⁽ᵍ²⁾  →  Γ ⊢ ℑ (A ⊗ B) ≃ ℑ A ⊗ ◯ B ⊕ ◯ A ⊗ ℑ B : Uᵢ⁽¹⁾ ⊕ Uᵢ⁽¹⁾\nΓ ⊢ 𝖘 (A ⊗ B) ≃ 𝖘 A ⊗ 𝖘 B : Uᵢ⁽ᵍ¹ + ᵍ²⁾\n```\n\nIdempodence:\n\n```\n◯ (◯ (𝖘 ℝ¹|¹)) ≃ ◯ (𝖘 ℝ¹|¹) ≃ ℝ¹.\nℑ (ℑ (𝖘 ℝ¹|¹)) ≃ ℑ (𝖘 ℝ¹|¹) ≃ ℝ⁰|¹.\n```\n\nProjections:\n\n```\nπ_◯ : 𝖘 ℝ¹|¹ → ◯ (𝖘 ℝ¹|¹) (x ↦ x).\nπ_ℑ : 𝖘 ℝ¹|¹ → ℑ (𝖘 ℝ¹|¹) (ψ ↦ ψ).\n```\n\nTensor:\n\n```\n◯ (𝖘 ℝ¹|¹ ⊗ 𝖘 ℝ¹|¹) ≃ ◯ (𝖘 ℝ¹|¹) ⊗ ◯ (𝖘 ℝ¹|¹) ≃ ℝ¹ ⊗ ℝ¹.\nℑ (𝖘 ℝ¹|¹ ⊗ 𝖘 ℝ¹|¹) ≃ ℝ⁰|¹ ⊗ ℝ¹ ⊕ ℝ¹ ⊗ ℝ⁰|¹ (two odd terms).\n```\n\nAdjuncion:\n\n```\nHom(◯ (𝖘 ℝ¹|¹), 𝖘 ℝ¹|¹) ≅ Hom(𝖘 ℝ¹|¹, ℑ (𝖘 ℝ¹|¹)) (e.g., maps ℝ¹ → ℝ¹|¹ vs. ℝ¹|¹ → ℝ⁰|¹).\nΓ ⊢ Hom(◯ (𝖘 A), 𝖘 B) ≅ Hom(𝖘 A, ℑ (𝖘 B))\nΓ ⊢ 𝖘 A : Uᵢ⁽ᵍ⁾ →  Γ ⊢ η : 𝖘 A → ℑ (𝖘 A)\nΓ ⊢ 𝖘 B : Uᵢ⁽ᵍ⁾ →  Γ ⊢ ε : ◯ (𝖘 B) → 𝖘 B\n```\n\n## TED-S (Supergeormetry, Felix) Examples\n\n* Graded Universes: ⊢ Uᵢ⁰ : Uᵢ₊₁⁰, ⊢ Uᵢ¹ : Uᵢ₊₁⁰.\n* Graded Tensor: Γ ⊢ A : Uᵢ^g₁, Γ ⊢ B : Uᵢ^g₂ → Γ ⊢ A ⊗ B : Uᵢ^(g₁ + g₂) Γ ⊢ a : A, Γ ⊢ b : B → Γ ⊢ a ⊗ b : A ⊗ B Γ ⊢ a : A^g₁, Γ ⊢ b : B^g₂ → Γ ⊢ a ⊗ b = (−1)^(g₁ g₂) b ⊗ a : A ⊗ B.\n* Group Action: Γ, g : 𝔾 ⊢ A : Uᵢ^g → Γ ⊢ 𝔾 → A : Uᵢ^g.\n* Super Type Theory: Uᵍᵢ`|` 𝖘 A `|` 𝔾 → A.\n* Super Modality: Γ ⊢ A : Uᵢ^g → Γ ⊢ 𝖘 A : Uᵢ^g.\n* Cohesive Type Theory: ♭ `|` ♯ `|` ℑ `|` ◯ (four built-in modalities).\n* Flat Codiscrete: Γ ⊢ A : Uᵢ^g → Γ ⊢ ♭ A : Uᵢ^g\n* Sharp Discrete:  Γ ⊢ A : Uᵢ^g → Γ ⊢ ♯ A : Uᵢ^g\n* Bosonic: Γ ⊢ A : Uᵢ⁽ᵍ⁾  →  Γ ⊢ ◯ A : Uᵢ⁽⁰⁾\n* Fermionic: Γ ⊢ A : Uᵢ⁽ᵍ⁾  →  Γ ⊢ ℑ A : Uᵢ⁽¹⁾\n\n∫ modality:\n\n```\nΓ ⊢ A : Uᵢ⁽ᵍ⁾  →  Γ ⊢ ∫ A : Uᵢ⁽ᵍ⁾,  ∫ A := Π (X : Uᵢ⁽ᵍ⁾), (♯ X → A) → ♭ X\nΓ ⊢ a : A  →  Γ ⊢ ∫ a : ∫ A,  ∫ a := η_∫ a,  η_∫ a := λ X. λ f. ♭ (f (η_♯ a))\nΓ ⊢ ε_♭ : ∫ (♭ A) → ♭ A\nΓ ⊢ Hom(∫ A, B) ≅ Hom(A, ♭ B)\nΓ ⊢ ∫ (∫ A) ≃ ∫ A\n```\n\n∇ modality:\n\n```\nΓ ⊢ A : Uᵢ⁽ᵍ⁾  →  Γ ⊢ ∇ A : Uᵢ⁽ᵍ⁾,  ∇ A := Σ (X : Uᵢ⁽ᵍ⁾), (A → ♭ X) × (♯ X → A)\nΓ ⊢ a : A  →  Γ ⊢ ∇ a : ∇ A,  ∇ a := η_∇ a,  η_∇ a := (𝟙, λ _ : ♭ 𝟙. ♭ a, λ x : ♯ 𝟙. a)\nΓ ⊢ η_♯ : ♯ A → A\nΓ ⊢ ε_∇ : ♯ (∇ A) → A\nΓ ⊢ ∇ (∇ A) ≃ ∇ A\n```\n\n𝖘 `ℝ¹ᴵ¹` lifts `ℝ¹ᴵ¹` to a super-context, expected to be isomorphic (𝖘 `ℝ¹ᴵ¹` ≃ `ℝ¹ᴵ¹`), as it’s already a supertype:\n\n```\nΓ ⊢ ℝ¹|¹ : U₀^|0| → Γ ⊢ 𝖘 ℝ¹|¹ : U₀^|0|.\n```\n\nEven part, ◯ (𝖘 `ℝ¹|¹`) ≃ ℝ¹ (bosonic coordinate space):\n\n```\nΓ ⊢ 𝖘 ℝ¹|¹ : U₀^|0| → Γ ⊢ ◯ (𝖘 ℝ¹|¹) : U₀^|0|.\n```\n\nOdd part, ℑ (𝖘 `ℝ¹|¹`) ≃ `ℝ⁰|¹` (fermionic coordinate):\n\n```\nΓ ⊢ 𝖘 ℝ¹|¹ : U₀^|0| → Γ ⊢ ℑ (𝖘 ℝ¹|¹) : U₀^|1|.\n```\n\nTensor Product:\n\n```\nΓ ⊢ ◯ (𝖘 ℝ¹|¹ ⊗ 𝖘 ℝ¹|¹) : U₀^|0| ≃ ℝ¹ ⊗ ℝ¹ (even part) \nℑ(𝖘 ℝ¹|¹) ⊗ ◯(𝖘 ℝ¹|¹) ⊕ ◯(𝖘 ℝ¹|¹) ⊗ ℑ(𝖘 ℝ¹|¹) ≃ ℝ⁰|¹ ⊗ ℝ¹ ⊕ ℝ¹ ⊗ ℝ⁰|¹ : U₀^|1| ⊕ U₀^|1|.\nΓ ⊢ θ₁ : ℝ^|1|, Γ ⊢ θ₂ : ℝ^|1| → Γ ⊢ θ₁ ⊗ θ₂ = −θ₂ ⊗ θ₁ : ℝ^|1| ⊗ ℝ^|1|\nΓ ⊢ θ : ℝ^|1| → Γ ⊢ θ ⊗ θ = 0 : ℝ^|1| ⊗ ℝ^|1|\nΓ ⊢ θ₁ : ℝ^|1|, Γ ⊢ θ₂ : ℝ^|1| → Γ ⊢ θ₁ ⊗ θ₂ : ℝ^|1| ⊗ ℝ^|1|\nΓ ⊢ θ₁ : ℝ^|1|, Γ ⊢ θ₂ : ℝ^|1| → Γ ⊢ θ₁ ⊗ θ₂ = (−1)^(|1| |1|) θ₂ ⊗ θ₁ : ℝ^|1| ⊗ ℝ^|1|\nΓ ⊢ θ₁ ⊗ θ₂ = −θ₂ ⊗ θ₁ : ℝ^|1| ⊗ ℝ^|1|\nΓ ⊢ θ₁ ⊗ θ₂ = (−1)^(|1| |1|) θ₂ ⊗ θ₁ = −θ₂ ⊗ θ₁ : ℝ^|1| ⊗\nΓ ⊢ ℝ^|0| : Uᵢ^|0|\nΓ ⊢ ℝ^|1| : Uᵢ^|1|\nΓ ⊢ ℝ^|0| ⊗ ℝ^|1| : Uᵢ^|1|\n|0| + |1| = |1|\nΓ ⊢ x : ℝ^|0| Γ ⊢ θ : ℝ^|1| Γ ⊢ x ⊗ θ : ℝ^|0| ⊗ ℝ^|1|\nΓ ⊢ g : G → Γ ⊢ t g : ℝ^|0| ⊗ ℝ^|1| ⊗ ℝ^|1|\n⊢ Γ, g : G, a : ℝ^|0| ⊗ ℝ^|1| ⊗ ℝ^|1| → Γ\n  ⊢ λg.a : G → ℝ^|0| ⊗ ℝ^|1| ⊗ ℝ^|1|\n  ⊢ t : G → ℝ^|0| ⊗ ℝ^|1| ⊗ ℝ^|1|\n```\n\n### TED-K (K-Theory, Jack) Examples\n\n* Stable Homotopy Primitives: Fib^n, Susp(A), Trunc^n, Π(x:A).B, Σ(x:A).B, Id_A(u,v), Spec, πₙ^S(A), S⁰[p], Group, A ∧ B, [A, B], Hⁿ(X; G), G ⊗ H, SS(E, r).\n* Cohesive Spectra: Linear types like H (Hilbert spaces),  PU(H), Fred^0(H).\n* Parameterized Spectra: X: Type ⊢ E(X):Spec, e.g., KU_G^τ(X;C).\n* Qubit Type: KU_G^τ(Config;C) := [Config,Fred^0(H)].\n* Modalities: ♭, ♯, ℑ, ◯, with ℑ(Config)≃BB_n (braid group delooping).\n* Key Feature: KU_G^τ encodes su(2)-anyonic ground states, with linear types for braiding.\n\nFibonacci Anions:\n\n```\ndef FibAnyon : Type_{lin} := 1 + τ\ndef FibState (c : Config) : Type_{lin} := Σ (a : FibAnyon), KUᶿ(c; ℂ)\ndef FibFusionRule : FibAnyon → FibAnyon → Type_{lin}\ndef FibFusionRule (1 a : FibAnyon) := Id_{FibAnyon}(a, a)\ndef FibFusionRule (τ τ : FibAnyon) := 1 + τ\ndef fuseFib (a b: FibAnyon) : Type := Σ (c: FibAnyon), FibFusionRule a b\ndef fuseFib (τ τ: FibAnyon) : Type ≡ (c, proof) where c : 1 + τ, proof : Id_{FibAnyon}(c, 1 + τ)\ndef fuseFibState (s₁ s₂ : FibState c) : FibState c := \\ (a₁, q₁) (a₂, q₂), (c, fuseQubit(q₁, q₂, c))\ndef measureFib : Σ(c : FibAnyon).FibFusionRule a b → FibState Config := (c, qubit_c)\n\ndef τ₁ : FibAnyon :≡ τ\ndef τ₂ : FibAnyon :≡ τ\ndef s₁ : FibState c :≡ (τ₁, q₁)\ndef s₂ : FibState c :≡ (τ₂, q₂)\ndef fused : Σ(c : FibAnyon), FibFusionRule τ τ :≡ fuseFib τ₁ τ₂\ndef resolved : FibState c :≡ measureFib fused\n```\n\nFusion for su(2) k-Anyonic States:\n\n```\ndef j₀ : Su2Anyon 2 :≡ 0\ndef j₁ : Su2Anyon 2 :≡ 1/2\ndef state : Su2State c 2 :≡ (j₁, qubit)\ndef braid 2 j₁ j₁ : KU^\\tau_G(c; ℂ)\ndef Su2Anyon : ℕ → Type_{lin}\ndef Su2Anyon k :≡ { j : ℝ | 0 ≤ j ≤ k/2 ∧ 2j ∈ ℕ }\ndef Su2FusionRule : ℕ → Su2Anyon k → Su2Anyon k → Type\ndef Su2FusionRule k j₁ j₂ ≡ Σ(j : Su2Anyon k).(|j₁ - j₂| ≤ j ≤ min(j₁ + j₂, k - j₁ - j₂))\ndef braid (k : ℕ) (a b : Su2Anyon k) : KU^\\tau_G(Config; ℂ)\ndef braid k a b :≡ R_{ab} · state(a, b)\ndef fuseSu2State (k : ℕ) (s₁ s₂ : Su2State c k) : Su2State c k\ndef fuseSu2State k (j₁, q₁) (j₂, q₂) :≡ (j, fuseQubit(q₁, q₂, j))\ndef fuse (k : ℕ) (j₁ j₂ : Su2Anyon k) := Σ(j : Su2Anyon k), Id_{Su2Anyon k}(j, fuseRule(j₁, j₂))\ndef fuse k j₁ j₂ :≡ (j, proof) where j ∈ {|j₁ - j₂|, ..., min(j₁ + j₂, k - j₁ - j₂)}\ndef fuseSu2 (k : ℕ)(j₁ j₂ : Su2Anyon k) : Su2FusionRule k j₁ j₂\ndef fuseSu2 k j₁ j₂ :≡ (j, proof)\n    where j = choose(|j₁ - j₂|, min(j₁ + j₂, k - j₁ - j₂)),\n    proof : Id_{Su2Anyon k}(j, fusionTerm(j₁, j₂))\n```\n\nMajorana Zero Modes:\n\n```\ndef MZM : Type_{lin} := γ\ndef MZMState (c: Config) : Type_{lin} := Σ(m : MZM), KU¹(c; ℂ)\ndef fuseMZM (m₁ m₂ : MZM) := Σ (c : FibAnyon), MZMFusionRule m₁ m₂\ndef fuseMZM (γ γ : MZM) ≡ (1, refl)\ndef resolveMZM : Σ(c : FibAnyon), MZMFusionRule γ γ → FibState Config\ndef resolveMZM (1, refl) ≡ (1, qubit_1)\ndef fuseMZMState (s₁ s₂ : MZMState c) : FibState c := \\ (γ, q₁) (γ, q₂), (1, fuseQubit(q₁, q₂, 1))\n```\n\n## Bibliography\n\n* Felix Cherubini. \u003ca href=\"https://d-nb.info/1138708615/34\"\u003eFormalizing Cartan Geometry in Modal Homotopy Type Theory\u003c/a\u003e. PhD.\n* Daniel R. Licata, Michael Shulman, Mitchell Riley. \u003ca href=\"https://github.com/mikeshulman/cohesivett\"\u003eA Fibrational Framework for Substructural and Modal Logics\u003c/a\u003e.\n* Branislav Jurco, Christian Sämann, Urs Schreiber, Martin Wolf. \u003ca href=\"https://arxiv.org/pdf/1903.02807\"\u003eHigher Structures in M-Theory\u003c/a\u003e.\n* Urs Schreiber. \u003ca href=\"https://arxiv.org/pdf/1310.7930\"\u003eDifferential cohomology in a cohesive ∞-topos\u003c/a\u003e.\n* John Huerta, Urs Schreiber. \u003ca href=\"https://arxiv.org/pdf/1702.01774\"\u003eM-theory from the Superpoint\u003c/a\u003e.\n* Namdak Tonpa, \u003ca href=\"https://tonpa.guru/stream/2018/2018-06-09 Cohesive Type Theory.htm\"\u003e2018-06-09 Cohesive Type Theory\u003c/a\u003e.\n* Namdak Tonpa, \u003ca href=\"https://tonpa.guru/stream/2019/2019-03-16 Geometry in Modal HoTT.htm\"\u003e2019-03-16 Geometry in Modal HoTT\u003c/a\u003e.\n* Namdak Tonpa, \u003ca href=\"https://tonpa.guru/stream/2020/2020-03-24 Modal Homotopy Type Theory.htm\"\u003e2020-03-24 Modal Homotopy Type Theory\u003c/a\u003e.\n* Namdak Tonpa, \u003ca href=\"https://tonpa.guru/stream/2023/2023-04-04%20%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80.htm\"\u003e2023-04-04 Суперпростір\u003c/a\u003e.\n* Namdak Tonpa, \u003ca href=\"https://urs.groupoid.space\"\u003eUrs: Equivariant Super Type Theory\u003c/a\u003e.\n* Kac, V. G. \u003ca href=\"https://core.ac.uk/download/pdf/81957395.pdf\"\u003e Lie Superalgebras\u003c/a\u003e.\n* Roček, M., Wadhwa, N. \u003ca href=\"https://arxiv.org/pdf/hep-th/0408188\"\u003e On Calabi-Yau Supermanifolds\u003c/a\u003e.\n* Cremonini, C. A., Grassi, P. A. \u003ca href=\"https://arxiv.org/pdf/2106.11786\"\u003e Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms\u003c/a\u003e.\n* Davis, S. \u003ca href=\"https://polipapers.upv.es/index.php/AGT/article/view/1623/1735\"\u003e Supersymmetry and the Hopf Fibration\u003c/a\u003e.\n* Aguilar, M. A., Lopez-Romero, J. M., Socolovsky, M. \u003ca href=\"https://inspirehep.net/files/72a57b4474bdb1f83e6963d1586050d0\"\u003eCohomology and Spectral Sequences in Gauge Theory\u003c/a\u003e.\n* Hisham Sati, Urs Schreiber. \u003ca href=\"https://arxiv.org/pdf/2209.08331\"\u003eTopological Quantum Programming in TED-K\u003c/a\u003e.\n\n## Author\n\n* Namdak Tonpa\n\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fgroupoid%2Furs","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fgroupoid%2Furs","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fgroupoid%2Furs/lists"}